Van  Antwerp,  0ragg  SlCo. 
Cincinnati  S.  NewYork. 


• - ^ - » 


THE  UNIVERSITY 


OF  ILLINOIS 
LIBRARY 

.  The 

Frank  Hall  collection 
of  arithmetics! 
presented  by  Professor 
Ht  L.  Rietz  of  the 
University  of  Iowa* 

51 3 
Ms 

miw&ms:. usrar*  ; 


/ 


I 


WHITE'S  GRADED-SCHOOL  SERIES 


A 

COMPLETE 

ARITHMETIC 

UNITING 


MENTAL  AND  WRITTEN  EXERCISES 


IN  A 


NATURAL  SYSTEM  OE  INSTRUCTION 


By  eCwHITE,  M.A.,  LL.D. 


YAN  ANTWERP,  BRAGG  &  CO. 


CINCINNATI 


NEW  YORK 


McGuffey’s  Revised  Readers  and  Speller. 
Me  Guffey’s  Revised  Charts. 

White’s  Arithmetics. 

Harvey's  Language  Course. 

Eclectic  Geographies. 

Eclectic  Penmanship. 

Eclectic  History  of  the  United  States. 
Thalheimer’s  Historical  Series. 


Entered  according  to  Act  of  Congress  in  the  year  1870,  by 
WILSON,  HINKLE  &  CO., 

In  the  Clerk’s  Office  of  the  District  Court  of  the  United  States  for 
the  Southern  District  of  Ohio. 


ELECTROTYPKt)  AT 
THE  FRANKLIN  TYPE  FOUNDRY, 
CINCINNATI. 


BLRCTIC  PRESS! 

VAN  ANTWERP,  BRAGG  &  CO,, 
CINCINNATI. 


&3Jc2.\  IAV. 


s$fc 

6>P'X 


wnmncs  uatuttj 


PREFACE. 


This  work  is  called  a  Complete  Arithmetic,  because  it  em¬ 


braces  all  the  subjects  which  properly  belong  to  a  school  arith 


metic.  It  is  designed  to  be  a  complete  text-book  for  pupils 
who  have  a  knowledge  of  the  fundamental  operations  with  in¬ 
tegral  numbers,  including  denominate  numbers. 

The  work  is  characterized  by  ttye  same  features  as  the  lower 
books  of  the  series,  viz.: 

1.  It  combines  Mental  and  Written  Arithmetic  in  a  practical  and 
philosophical  manner.  This  is  done  by  making  the  mental  exer¬ 
cises  preparatory  to  the  written ;  and  thus  these  two  classes  of 
exercises,  which  have  been  so  long  and  so  unnaturally  divorced, 
are  united  as  the  essential  complements  of  each  other. 

2.  It  faithfully  embodies  the  inductive  method  of  teaching.  The 
wTritten  methods  are  preceded  by  the  analysis  of  mental  problems, 
and  both  the  written  methods  and  the  principles  which  they  in¬ 
volve,  are  derived  inductively  from  the  analytic  processes.  The 
successive  steps  of  each  process  are  mastered  by  the  pupil 
through  the  solution  of  problems,  and  he  is  required  to  deduce 
and  state  the  rules  before  he  is  confronted  with  the  author’s 
generalizations.  All  definitions  which  are  deducible  from  the 
processes,  and,  with  few  exceptions,  all  principles  and  rules,  are 
placed  after  the  problems — a  feature  peculiar  to  this  Series. 

3.  It  is  specially  adapted ,  both  in  matter  and  method,  to  the  grade  of 
pupils  for  which  it  is  designed.  The  greater  portion  of  the  work 
is  devoted  to  a  progressive  and  thorough  treatment  of  subjects 
not  embraced  in  the  lower  books  —  an  arrangement  which  spe- 


(iii) 


IV 


PREFACE. 


cially  meets  the  wants  of  Graded  Schools.  Not  more  than  twenty 
pages  are  in  any  sense  a  repetition.  The  repeated  matter  con¬ 
sists  of  definitions,  principles,  and  rules,  all  the  problems  being 
new.  The  subjects  before  treated  are  not  only  concisely  reviewed, 
but  from  a  higher  stand-point.  Of  the  twenty-four  pages  devoted 
to  the  fundamental  rules,  eight  present  new  abbreviated  methods; 
and  of  twenty-eight  pages  devoted  to  Denominate  Numbers,  sim¬ 
ple  and  compound,  more  than  sixteen  discuss  new  topics.  A  sim¬ 
ilar  difference  is  observable  in  the  treatment  of  Common  Fractions, 
Decimal  Fractions,  United  States  Money,  etc.  Among  the  added 
articles  worthy  of  special  mention  are  those  on  Denominate  Frac¬ 
tions,  the  Metric  System,  Longitude  and  Time,  and  Foreign  Ex¬ 
change. 

In  the  number  of  problems,  tbe  author  lias  aimed  to  bit  the 
golden  mean  between  a  paucity  and  an  excess,  and  the  greatest 
pains  has  been  taken  to  make  them  sufficiently  progressive, 
varied,  and  difficult,  to  afford  the  requisite  drill  and  practice. 
Instead  of  rehashing  old  problems,  with  their  incorrect  data  and 
obsolete  terms,  the  author  has  gone  to  science  and  history  for 
statistical  information  of  practical  value,  and  he  has  aimed  to 
present  the  current  values,  terms,  forms,  and  usages  of  American 
business.  The  mental  problems  will  be  found  as  difficult  and 
comprehensive  as  those  which  constitute  the  latter  half  of  the 
standard  Mental  Arithmetics,  and  are  sufficiently  numerous  to 
afford  thorough  drills  in  analysis. 

The  explanations  of  the  written  processes  are  not  designed  to 
serve  as  models  for  the  pupil  to  memorize  and  repeat.  They  are 
intended  to  supplement  the  analysis.  In  some  cases,  a  formal 
analysis  is  given ;  in  others,  a  principle  is  deduced  or  demon¬ 
strated  ;  and  in  others,  the  process  is  described  or  its  principles 
stated.  Neither  teacher  nor  pupil  is  denied  the  privilege  of  de¬ 
termining  his  own  explanations. 

Another  characteristic  feature  of  this  work  is  the  prominence 
given  to  Principles.  A  clear  comprehension  of  the  principles 


PREFACE. 


v 


of  arithmetic  is  essential  to  its  thorough  mastery,  and  their  in¬ 
duction,  proof,  and  illustration  are  mental  exercises  of  great 
value.  Until  the  pupil  can  step  inductively  from  processes  to 
principles,  he  has  not  a  thorough  knowledge  of  numbers.  In  this 
work  the  principles  are  concisely  and  formally  stated  in  connec¬ 
tion  with  the  rules  which  are  based  upon  them. 

The  author  invites  special  attention  to  the  treatment  of  Per¬ 
centage.  Over  eighty  pages  are  devoted  to  this  subject  and  its 
applications,  and  it  is  believed  that  the  treatment  will  be  found 
not  only  full  and  thorough,  but  of  great  practical  value.  The 
student  who  masters  these  pages  will  certainly  have  a  fair 
knowledge  of  the  nature,  laws,  and  usages  of  the  business  of  the 
country.  The  introduction  of  Formulas ,  it  is  hoped,  will  prove  a 
useful  feature. 

The  thorough  treatment  of  Ratio  before  Proportion,  and  of  the 
latter  before  its  application  to  the  solution  of  problems,  will  make 
the  mastery  of  this  subject  easy.  The  treatment  of  Involution 
and  Evolution  will  not  escape  notice.  The  geometrical  explana¬ 
tions  of  Square  Root  and  Cube  Root  are  the  reverse  of  those 
usually  given,  and  are  believed  to  be  new.  They  will  be  found 
both  simple  and  natural. 

The  Complete  Arithmetic  is  submitted  to  American  teach¬ 
ers  in  the  hope  that  it  may  not  only  be  found  new  in  its  general 
plans  and  in  many  of  its  methods  and  details,  but  that  it  may 
prove  eminently  adapted  to  the  present  wants  and  condition  of 
Graded  Systems  of  Instruction. 

Columbus,  Ohto,  July,  1870. 

N.  B. — The  steadily  increasing  use  of  this  Three-book  Series 
of  Arithmetics  has,  from  time  to  time,  demanded  not  only  the 
correction  of  all  discovered  errors,  but  such  other  slight  revisions 
as  have  been  made  necessary  by  changes  in  business  usages, 
laws,  and  values.  This  edition  is  believed  to  be  fully  up  with 
the  present  condition  of  business.  -  ■ 

Cincinnati,  Ohio,  Jan.  10,  1884. 


SUGGESTIONS  TO  TEACHERS. 

/ 

.  1.  The  Mental  Problems  should  be  made  a  thorough  drill  in  analy* 
sis;  but,  since  the  reasoning  faculty  is  not  trained  by  mere  logical 
verbiage,  the  solution  should  be  concise  and  simple.  They  should 
also  be  made  introductory  to  the  written  processes  of  which  they  are 
often  a  complete  elucidation.  Many  of  the  written  problems  may 
also  be  solved  mentally,  thus  increasing  the  drills  in  analysis. 

2.  All  Written  Problems  should  be  solved  by  the  pupils  on  slate  or 
paper,  and  the  solutions  should  be  brought  to  the  recitation  for  the 
teacher’s  inspection  and  criticism.  From  three  to  five  minutes  at  the 
beginning  of  the  recitation  will  suffice  to  ascertain  the  accuracy  and 
neatness  of  each  pupil’s  work.  The  explanations  of  the  written  pro¬ 
cesses,  given  by  the  pupil,  should  be  both  analytic  and  inductive.  A 
part  of  the  mental  problems  should  also  be  solved  as  written  prob¬ 
lems,  thus  making  the  induction  of  the  written  process  easy. 

3.  The  Definitions  should  be  deduced  and  stated  by  the  pupils  under 
the  guidance  of  the  teacher,  and  this  can  usually  be  done  in  connec¬ 
tion  with  the  solution  of  the  problems.  See  Int.  Arith.,  p.  5,  Sug.  3. 
When  the  definitions  are  placed  before  the  problems,  as  in  the  appli¬ 
cations  of  Percentage,  they  should  be  studied  by  the  pupils,  but  their 
recitation  may  be  deferred  until  the  problems  are  solved,  and  the 
processes  mastered. 

4.  The  Principles  should  be  taught  inductively,  when  this  is  possi¬ 
ble,  and  each  should  be  proved  or  illustrated,  or  both,  by  the  pupil. 
A  thorough  mastery  of  every  principle  should  be  made  an  essential 
condition  of  the  pupil’s  progress.  The  recitation  should  secure  a  con¬ 
stant  application  of  known  principles,  and  a  clear  comprehension  of 
all  new  ones. 

5.  The  Rules  should  also  be  deduced  and  stated  by  the  pupils. 
The  true  order  is  this:  1.  A  mastery  of  the  process.  2.  Recognition 
of  the  successive  steps  in  order,  and  a  statement  of  each.  3.  Combi¬ 
nation  of  these  several  statements  into  a  general  statement.  4.  Com¬ 
parison  of  this  generalization  with  the  author’s  rule.  5.  Memorizing 
of  the  rule  approved.  See  Int.  Arith.,  p.  6. 

6.  When  two  or  more  methods  or  solutions  are  given,  the  one  pre¬ 
ferred  should  be  thoroughly  taught.  It  is  well  for  pupils  to  understand 
different  processes  and  explanations,  but  they  should  be  made 
familiar  with  one  of  them. 

7.  Before  a  subject  is  left,  the  pupils  should  be  required  to  make  a 
topical  analysis  of  the  definitions,  principles,  and  rules,  and  the  same 
should  be  recited  with  accuracy  and  dispatch. 

N.  B. —  See  the  author’s  “Manual  of  Arithmetic”  for  other  sugges¬ 
tions,  methods  of  teaching,  models  of  analysis,  illustrative  solutions, 
etc. 


CONTENTS 


SECTION  I -VI. — The  Fundamental  Rules. 


PAGE 

Notation  and  Numeration  .  10 

Addition . 13 

The  Addition  of  Two  Columns  .  15 

Subtraction  .  .  .  .17 


Multiplication  . 
Abbreviated  Processes 
Division 

Abbreviated  Processes 


SECTION  VII. — Properties  op  Numbers. 

Divisors  and  Factors  .  .  32  Greatest  Common  Divisor  . 

Cancellation  .  .  .  .35  Least  Common  Multiple 


SECTION  VIII.— Fractions. 


Notation  and  Numeration 
Reduction  of  Fractions 
Addition  of  Fractions 
Subtraction  of  Fractions  . 
Multiplication  of  Fractions 


Division  of  Fractions  . 
Complex  Fractions 
Numbers  Parts  of  Other  Num 
bers  .... 
Review  of  Fractions 


.  43 
.  46 
.  53 
.  55 
.  57 


SECTION  IX. — Decimal  Fractions. 


Numeration  and  Notation 
Reduction  of  Decimals 
Addition  of  Decimals 


73 

79 

82 


Subtraction  of  Decimals  . 
Multiplication  of  Decimals 
Division  of  Decimals 


SECTION  X. — United  States  Money. 


Notation  and  Reduction  . 
Addition  and  Subtraction  . 
Multiplication  and  Division 


90 

91 

92 


Abbreviated  Methods 
Aliquot  Parts 
Bills  . 


SECTION  XI. — Mensuration. 
Surfaces . 100  |  Solids  . 


SECTION  XII. — Denominate  Numbers. 


Reduction . 107 

Denominate  Integers  and  Mixed 
Numbers  ....  107 


Denominate  Fractions 
The  Metric  System  . 
Metric  Tables 


(vii) 


PAGE 

20 

22 

25 

28 


37 

40 


61 

66 

66 

68 


83 

84 

85 


93 

95 

97 


104 


110 

119 

120 


CONTENTS, 


•  •  • 
Vlll 


SECTION  XIII.— Compound  Numbers. 


PAGE 

PAGE 

Addition  and  Subtraction  . 

125 

Longitude  and  Time 

• 

• 

130 

Multiplication  and  Division 

128 

SECTION  XIV. 

—Percentage. 

The  Four  Cases  of  Percentage  . 

137 

Six  Per  Cent  Method 

• 

175 

Review  of  the  Four  Cases  . 

144 

Method  by  Days 

• 

179 

Applications  of  Percentage  . 

146 

Partial  Payments  . 

• 

181 

Profit  and  Loss ...» 

146 

The  Problems  in  Interest 

185 

Commission  and  Brokerage  . 

149 

Review  of  Problems 

a 

190 

Capital  and  Stock  . 

154 

Present  Worth  and  Discount 

192 

Insurance . 

159 

Bank  Discount 

• 

194 

Life  Insurance .... 

163 

Notes,  Drafts,  and  Bonds 

• 

198 

Taxes . 

164 

Exchange 

• 

201 

Customs  or  Duties  . 

168 

Annual  Interest  . 

• 

205 

Bankruptcy  . 

170 

Compound  Interest 

• 

208 

Interest  . 

171 

Equation  of  Payments , 

• 

211 

General  Method 

172 

Equation  of  Accounts  . 

• 

215 

SECTION  XV. 

—Ratio  and  Proportion. 

Patio ....... 

220 

Partnership 

• 

• 

234 

Proportion . 

224 

Simple  Partnership 

» 

• 

235 

Simple  Proportion 

225 

Compound  Partnership 

• 

• 

237 

Compound  Proportion . 

230 

Problems  for  Analysis  . 

• 

• 

239 

SECTION  XVI.- 

Involution  and  Evolution. 

Involution . 

246 

Geometrical  Explanation 

• 

• 

255 

Another  Method  of  Involution  . 

248 

Cube  Root 

• 

• 

257 

Evolution . 

249 

Geometrical  Explanation 

• 

• 

262 

Square  Root  .... 

251 

Mensuration,  involving  Inv.  &  Ev. 

264 

SECTION  XVII. 

General  Review. 

Test  Problems  .... 

268  |  Test  Questions 

• 

• 

281 

APPENDIX. 

Notation . 

287 

Geometrical  Progression 

.  300 

Proofs  by  Excess  of  9’s  . 

287 

Alligation  .... 

.  303 

Circulating  Decimals  . 

289 

Duodecimals  .... 

.  306 

Tables  of  Denominate  Numbers 

291 

Permutations  .... 

.  308 

Legal  Rates  of  Interest . 

295 

Annuities  .... 

.  309 

Life  Insurance  .  .  .  . 

295 

Rules  of  Mensuration  . 

.  309 

Equation  of  Payments . 

296 

Foreign  Exchange 

.  312 

Arithmetical  Progression  . 

297 

Answers . 

.  317 

COMPLETE  ARITHMETIC. 


SECTION  I. 

PRELIMINARY  DEFINITIONS. 

Art.  1.  Arithmetic  is  the  science  of  numbers,  and  the 
art  of  numerical  computation. 

As  a  science,  Arithmetic  treats  of  the  relations,  properties,  and 
principles  of  numbers;  and,  as  an  art ,  it  applies  the  science  of 
numbers  to  their  computation. 

2.  A  Unit  is  one  thing  of  any  kind. 

3.  A  Number  is  a  unit  or  a  collection  of  units. 

4.  An  Integer  is  a  number  composed  of  whole  or  inte¬ 
gral  units ;  as,  5,  12,  20.  It  is  also  called  a  Whole  Number. 

5.  Numbers  are  either  Concrete  or  Abstract. 

A  Concrete  Number  is  applied  to  a  particular  thing 
or  quantity ;  as,  4  stars,  6  hours. 

"When  a  concrete  number  expresses  the  denominate  units  of  cur¬ 
rency,  weight,  or  measure,  it  is  called  a  Denominate  Number.  (Art.  174.) 

An  Abstract  Number  is  not  applied  to  a  particular 
thing  or  quantity ;  as,  4,  6,  20. 

A  concrete  number  is  composed  of  concrete  units,  and  an  abstract 
number  of  abstract  units. 


(9) 


10 


COMPLETE  ARITHMETIC. 


6.  A  Problem  is  a  question  proposed  for  solution. 

7.  An  Example  is  a  problem  used  to  illustrate  a  process 
or  a  principle. 

8.  A  Pule  is  a  general  description  of  a  process. 

9.  An  Arithmetical  Sign  is  a  character  denoting  an 
operation  to  be  performed  with  numbers,  or  a  relation  between 
them. 

10.  In  the  Mental  Solution  of  a  problem,  the  suc¬ 
cessive  steps  are  determined  mentally,  and  the  results  are 
not  written. 

In  the  Written  Solution  of  a  problem,  the  results  are 
written  on  a  slate,  paper,  or  other  substance. 


SECTION  II. 

NOTATION  AND  NUMERATION. 

MENTAL  EXERCISES. 

1.  How  many  hundreds,  tens,  and  units  in  368?  427? 
549?  608?  724?  806?  870? 

2.  How  many  hundred-thousands,  ten-thousands,  and 
thousands  in  456048  ?  607803  ?  680435  ?  700450  ? 

3.  Read  the  thousands’  period  in  3045;  40607;  150482; 
405360;  920400;  600060. 

4.  Read  first  the  thousands’  period  and  then  the  units’ 
period  in  65671 ;  120408  ;  400750 ;  650400  ;  80008. 

5.  Read  45037406;  520600480 ;  138405050. 

6.  Read  50008140;  600650508;  805000030. 

7.  Read  5308008450;  35006060600;  120500408080. 

8.  Read  7008360004;  302000860060;  500080800008. 


NUMERATION  AND  NOTATION. 


11 


WRITTEN  EXERCISES. 

9.  Express  in  figures  the  number  composed  of  5  thou¬ 
sands,  7  tens,  and  3  units;  4  ten-thousands,  6  hundreds, 
and  5  units. 

10.  Express  in  figures  50  thousands  and  40  units;  406 
thousands  and  30  units;  700  thousands  and  7  units. 

Express  the  following  numbers  in  figures: 

11.  Five  million  five  thousand  five  hundred. 

12.  Sixty  million  sixty  thousand  and  sixty. 

13.  Seven  hundred  million  seven'  hundred  thousand  seven 
hundred. 

14.  Five  hundred  and  sixty  million  sixty-eight  thousand. 

15.  Four  billion  fourteen  million  forty-five  thousand. 

16.  Sixty-five  billion  six  thousand  and  fifty. 

17.  Three  hundred  and  fifty  billion  forty-nine  million. 

18.  Seventeen  trillion  seventy  billion  seven  hundred 
thousand  four  hundred. 

19.  Fifty-six  trillion  sixteen  million  and  ninety. 

20.  Seven  quadrillion  eighty-five  billion  two  hundred  and 
four. 


DEFINITIONS  AND  PRINCIPLES. 

11.  There  are  three  methods  of  expressing  numbers : 

1.  By  words;  as,  five,  fifty,  etc. 

2.  By  letters ,  called  the  Roman  method. 

3.  By  figures ,  called  the  Arabic  method. 

12.  Notation  is  the  art  of  expressing  numbers  by  fig¬ 
ures  or  letters. 

13.  Numeration  is  the  art  of  reading  numbers  ex¬ 
pressed  by  figures  or  letters. 

Note. — Notation  may  be  defined  to  be  the  art  of  writing  numbers, 
and  Numeration,  the  art  of  reading  numbers.  In  Arithmetic,  the  term 
notation  is  used  to  denote  the  Arabic  method. 

14.  In  the  Roman  Notation,  numbers  are  expressed  by 
means  of  seven  capital  letters,  viz:  I,  V,  X,  L,  C,  D,  M. 


12 


COMPLETE  ARITHMETIC. 


I  stands  for  one ;  V,  for  five ;  X,  for  ten ;  L,  for  fifty ;  C, 
for  one  hundred ;  D,  for  five  hundred ;  M,  for  one  thousand. 
All  other  numbers  are  expressed  by  repeating  or  combining 
these  letters.  A  bar  over  a  letter,  as  D,  M,  multiplies  its 
value  by  one  thousand. 

15.  In  the  Arabic  Notation,  numbers  are  expressed  by 
means  of  ten  characters,  called  figures;  viz.,  0,  1,  2,  3,  4, 
5,  6,  7,  8,  9. 

The  first  of  these  characters,  0,  is  called  Naughty  or 
Cipher.  It  denotes  nothing,  or  the  absence  of  number. 

The  other  nine  characters  are  called  Significant  Figures, 
or  Numeral  Figures.  They  each  express  one  or  more  units. 
They  are  also  called  Digits. 

16.  The  successive  figures  which  express  a  number,  denote 
successive  Orders  of  Units.  A  figure  in  units’  place  denotes 
units  of  the  first  order;  in  tens’  place,  units  of  the  second  order; 
in  hundreds’  place,  units  of  the  third  order,  and  so  on — the 
term  units  being  used  to  express  ones  of  any  order. 

17.  Figures  have  two  values,  called  Simple  and  Local. 

The  Simple  Value  of  a  figure  is  its  value  when  stand¬ 
ing  in  unit’s  place.  It  is  also  called  its  Absolute  value. 

The  Local  Value  of  a  figure  is  its  value  arising  from 
the  order  in  which  it  stands.  The  local  value  of  a  figure  is 
tenfold  greater  in  hundreds’  order  than  in  tens’  order. 

18.  The  local  value  of  each  of  the  successive  figures 
which  express  a  number,  is  called  a  Term.  The  terms  of 
325  are  3  hundreds,  2  tens,  and  5  units. 

19.  The  figures  denoting  the  successive  orders  of  units 
are  divided  into  groups  of  three  figures  each,  called  Periods. 
The  first  or  right-hand  period  is  called  Units;  the  second, 
Thousands;  the  third,  Millions;  the  fourth,  Billions;  the  fifth, 
Trillions;  the  sixth,  Quadrillions;  the  seventh,  Quintillions ; 
the  eighth,  Sextillions;  the  ninth,  Septillions;  the  tenth,  Oc¬ 
tillions;  the  eleventh,  Nonillions;  the  twelfth,  Decillions,  etc. 

Note. — The  division  of  orders  into  periods  of  three  figures  each  is 
the  French  method.  In  the  English  method,  the  period  contains  six 


ADDITION. 


13 


orders,  the  name  of  the  first  period  being  Units,  the  second  Millions, 
the  third  Billions,  etc. 

20.  The  three  orders  of  any  period,  counting  from  the 
right,  denote  respectively  Units,  Tens,  and  Hundreds  of  that 
period.  They  may  be  briefly  read  by  calling  the  first  order 
by  the  name  of  the  period,  and  uniting  the  words  ten  and 
hundred  in  each  period  after  the  first  with  the  period’s  name. 

Thus,  the  orders  of  thousands’  period  may  be  read  thou¬ 
sands,  ten-thousands,  hundred-thousands;  the  orders  of  millions’ 
period,  millions,  ten-millions,  hundred-millions,  etc. 

21.  Principles. — 1.  Ten  units  of  the  first  order  make 
one  unit  of  the  second  order,  ten  units  of  the  second  order 
make  one  unit  of  the  third,  and,  generally,  ten  units  of  any 
order  make  one  unit  of  the  next  higher  order.  Hence, 

2.  The  value  of  the  successive  orders  of  figures  increases  ten¬ 
fold  from  right  to  left. 

3.  The  value  of  a  figure  is  multiplied  by  10  by  each  removal 
of  it  one  order  to  the  left,  and  is  divided  by  10  by  each  removal 
of  it  one  order  to  the  right. 


SECTION  III. 

ADDITION. 


MENTAL  PROBLEMS. 

1.  Add  by  6’s  from  1  to  73,  thus:  1,  7,  13,  19,  25,  etc. 

2.  Add  by  7’s  from  3  to  73;  from  6  to  90. 

3.  Add  by  8’s  from  5  to  77 ;  from  7  to  95. 

4.  Add  by  9’s  from  4  to  76;  from  8  to  98. 

5.  The  ages  of  five  boys  are  respectively  12,  10,  9,  8,  and 
7  years:  what  is  the  sum  of  their  ages? 

6.  A  rode  45  miles  the  first  day,  42  miles  the  second  day, 
and  38  miles  the  third  day:  how  far  did  he  ride  in  all? 

Suggestion. — Add  the  tens  and  then  the  units  of  each  couplet,  thus: 
45  40  =  85,  85  +  2  =  87 ;  87  +  30  =  117, 117  +  8  =  125.  Or  name 

only  results,  thus:  45,  85,  87;  117,  125.  (Art.  22.) 


14 


COMPLETE  ARITHMETIC. 


7.  A  drover  bought  37  sheep  of  one  farmer,  44  sheep  of 
another,  48  sheep  of  another,  and  27  sheep  of  another:  how 
many  sheep  did  he  buy? 

8.  The  Senior  class  of  a  college  contains  27  students,  the 
Junior  class  34,  the  Sophomore  class  38,  and  the  Freshman 
class  46:  how  many  students  in  the  college? 

9.  A  grocer  sold  18  sacks  of  flour  on  Monday,  23  on 
Tuesday,  27  on  Wednesday,  24  on  Thursday,  35  on  Friday, 
and  37  on  Saturday:  how  many  sacks  did  he  sell  during  the 
week? 

10.  A  lady  paid  $36  for  a  carpet,  $34  for  a  bureau,  $16 
for  a  washstand,  $28  for  a  bedstead,  and  $42  for  chairs:  how 
much  did  she  pay  for  all? 

11.  A  man  paid  $85  for  a  horse,  and  $17  for  his  keeping; 
and  then  sold  him  so  as  to  gain  $15:  for  how  much  did  he 
sell  the  horse? 

12.  Two  men  start  from  the  same  point,  and  travel  in 
opposite  directions,  the  one  at  the  rate  of  54  miles  a  day, 
and  the  other  at  the  rate  of  48  miles  a  day:  how  far  will 
they  be  apart  at  the  close  of  the  second  day? 

WRITTEN  PROBLEMS. 

13.  Add  347,  4086,  7080,  29408,  and  67736. 

14.  667  +  3804  -f  45608  +  304867  +  87609  =  what? 

15.  Add  four  thousand  and  fifty-six ;  sixty-three  thousand 
seven  hundred ;  seven  million  nine  thousand  and  ninety-nine ; 
and  fifty-six  million  nine  hundred  and  seventy-eight. 

16.  Add  eight  million  eighty  thousand  eight  hundred; 
seven  hundred  thousand  and  seventy;  five  million  eighty-six 
thousand  seven  hundred  and  eight;  and  sixty  million  six 
hundred  thousand  and  seventy. 

17.  A  grain  dealer  bought  wheat  as  follows :  Monday, 
2480  bushels;  Tuesday,  788  bushels;  Wednesday,  1565 
bushels;  Thursday,  2684  bushels;  Friday,  985  bushels;  Sat¬ 
urday,  3867  bushels.  How  many  bushels  did  he  buy  during 
the  week? 


ADDITION. 


15 


18.  Ohio  contains  39964  square  miles;  Indiana,  33809; 
Illinois,  55409;  Michigan,  56243;  Wisconsin,  53924;  Min¬ 
nesota,  83531 ;  Iowa,  55045 ;  and  Missouri,  65350.  What  is 
the  total  area  of  these  eight  States  ? 

19.  The  population  of  these  States  in  1860  was  as  follows: 
Ohio,  2339511 ;  Indiana,  1350428;  Illinois,  1711951 ;  Michi¬ 
gan,  756890;  Wisconsin,  778714;  Minnesota,  189923;  Iowa, 
674913 ;  Missouri,  1182012.  What  was  their  total  popu¬ 
lation  ? 

20.  The  territory  of  the  United  States  has  been  acquired 
as  follows : 


Territory  ceded  by  England,  1783, 

Louisiana,  as  acquired  from  France,  1803, 

Florida,  as  acquired  from  Spain,  1821, 

Texas,  as  admitted  to  the  Union,  1845, 

Oregon,  as  settled  by  treaty,  1846, 

California,  etc.,  as  conquered  from  Mexico,  1847, 
Arizona,  as  acquired  from  Mexico  by  treaty,  1854, 
Alaska,  as  acquired  from  Kussia  by  purchase,  1867, 


Square  miles. 

815615 
.  930928 
59268 
.  237504 
280425 
.  649762 
27500 
.  577390 


What  is  the  total  area  of  the  United  States? 


ADDITION  OF  TWO  COLUMNS. 


22.  There  is  a  practical  advantage  in  adding  two  columns 
at  one  operation.  Some  accountants  add  three  oF more  col¬ 
umns  in  this  manner. 

21.  Add  67,  58,  43,  36,  and  54. 


Process. 


67 

Add  thus  :  54  -f 

o 

CO 

84,  +  6  = 

90; 

90  -f  40 

-  130,  + 

58 

3  ==  133 ;  133  + 

50  = 

183,  +  8  = 

=  191 ;  191  + 

60  =  251, 

43 

+  7  =  258. 

36 

Or  thus,  naming 

only  results:  54, 

84, 

90;  130, 

133,  183; 

54 

191 ;  251,  258. 

258 


Note. — The  process  consists  in  first  adding  the  tens  of  each  couplet, 
and  then  the  units.  If  preferred,  the  units  may  first  be  added,  and 
then  the  tens.  Sufficient  practice  will  enable  the  accountant  to  add 
two  columns  without  separating  the  numbers  into  tens  and  units. 


16 


COMPLETE  ARITHMETIC. 


22.  Add  37,  40,  63,  84,  67,  22,  and  70. 

23.  Add  95,  46,  77,  66,  88,  63,  33,  and  44. 

24.  Add  67,  76,  45,  54,  38,  83,  27,  and  72. 

25.  Add  68,  86,  97,  79,  86,  68,  78,  and  87. 

26.  Add  45,  60,  57,  86,  83,  76,  49,  58,  and  84. 

27.  Add  56,  75,  83,  96,  69,  73,  37,  38,  and  205. 

28.  Add  27,  72,  33,  38,  69,  96,  75,  57,  and  336. 

29.  Add  235,  88,  77,  66,  55,  44,  33,  22,  and  11. 

30.  Add  405,  56,  43,  47,  74,  36,  63,  75,  and  66. 

31.  Add  46,  67,  72,  38,  99,  87,  65,  74,  and  88. 

32.  Add  73,  86,  47,  56,  69,  65,  58,  33,  52,  and  94. 

DEFINITIONS  AND  PRINCIPLES. 

23.  Addition  is  the  process  of  finding  the  sum  of  two 
or  more  numbers. 

The  Sum  of  two  or  more  numbers  is  a  number  contain¬ 
ing  as  many  units  as  all  of  them,  taken  together.  It  is  also 
called  the  Amount. 

24.  The  Sign  of  Addition  is  a  short  vertical  line  bi¬ 
secting  an  equal  horizontal  line,  -{-.  It  is  called  'plus. 

25.  The  Sign  of  Equality  is  two  short  horizontal 
parallel  lines,  =.  It  is  read  equals  or  is  equal  to.  Thus, 
7  -f-  8  =  15  is  read  7  plus  8  equals  15. 

26.  Lilze  Numbers  are  composed  of  units  of  the  same 
kind.  Thus,  4  balls  and  8  balls,  or  4  dimes  and  8  dimes,  or 
4  and  8,  are  like  numbers. 

27.  Principles. — 1.  Only  like  numbers  can  be  added. 

2.  Only  like  orders  of  figures  can  be  added. 

3.  The  sum  is  of  the  same  kind  or  order  as  the  numbers  added. 

4.  The  sum  is  the  same  whatever  be  the  order  in  which  the 
numbers  are  added. 

Note. — See  appendix  for  method  of  proof  by  “casting  out  the  9’s.” 


SUBTRACTION. 


17 


SECTION  IY. 

SUBTRACTION. 


MENTAL  PROBLEMS. 

1.  Count  by  4’s  from  61  back  to  1,  thus :  61,  57,  53,  etc. 

2.  Count  by  6’s  from  53  back  to  5 ;  from  74  back  to  2. 

3.  Count  by  7’s  from  66  back  to  3 ;  from  85  back  to  1. 

4.  Count  by  8’s  from  75  back  to  3 ;  from  94  back  to  6. 

5.  Count  by  9’s  from  73  back  to  1 ;  from  96  back  to  6. 

6.  A  grocer  having  a  certain  number  of  sacks  of  flour, 
bought  48  sacks,  and  sold  33  sacks,  and  then  had  34  sacks 
on  hand:  how  many  sacks  had  he  at  first? 

7.  A  man  sold  a  horse  for  $95,  which  was  $28  more  than 
the  horse  cost  him :  what  was  the  cost  of  the  horse  ? 

8.  Two  men  start  at  once  from  the  same  point,  and  travel 
in  the  same  direction,  one  traveling  52  miles  a  day,  and  the 
other  but  39  miles :  how  far  will  they  be  apart  at  the  close 
of  the  second  day? 

9.  A  man  earns  $85  a  month,  and  pays  $18  for  house 
rent,  and  $35  for  other  expenses :  how  much  does  he  save 
each  month? 

10.  A  gentleman  being  asked  his  age  said,  that  if  he 
should  live  27  years  longer,  he  should  then  be  three  score 
and  ten  :  what  was  his  age  ? 

11.  From  a  piece  of  carpeting  containing  68  yards,  a  mer¬ 
chant  sold  27  yards  to  one  man  and  18  yards  to  another: 
how  many  yards  of  the  piece  were  left? 

12.  A  man  bought  a  carriage  for  $135,  paid  $21  for  re¬ 
pairing  it,  and  then  sold  it  for  $170:  how  much  did  he  gain? 

13.  A  boy  earned  65  cents,  and  his  father  gave  him  33 
cents;  he  paid  45  cents  for  an  arithmetic  and  18  cents  for  a 
slate:  how  much  money  had  he  left? 

14.  There  are  85  sheep  in  three  fields;  there  are  36  sheep 

C.Ar. — 2. 


18 


COMPLETE  ARITHMETIC. 


in  the  first  field,  and  28  sheep  in  the  second :  how  many  sheep 
in  the  third  field? 

15.  John  had  33  chestnuts,  and  Charles  25;  John  gave 
Charles  14  chestnuts,  and  Charles  gave  his  sister  as  many  as 
he  then  had  more  than  John:  how  many  chestnuts  did  the 
sister  receive? 


WRITTEN  PROBLEMS. 

16.  A  builder  contracted  to  build  a  school-house  for  $25460, 
and  the  job  cost  him  $21385:  what  were  his  profits? 

17.  The  earth’s  mean  distance  from  the  sun  (old  value) 
is  95274000  miles,  and  that  of  Mars  is  145168136:  how 
much  farther  is  Mars  from  the  sun  than  the  earth? 

18.  The  population  of  Illinois  in  1860  was  1711951,  and 
in  1865  its  population  was  2141510 :  what  was  the  increase 
in  five  years? 

19.  The  population  of  Massachusetts  in  1860  was  1231066, 
and  in  1865  it  was  1267031 :  what  was  the  increase  in  five 
years  ? 

20.  The  area  of  the  Chinese  Empire  is  4695334  square 
miles,  and  the  area  of  the  United  States  is  3578392  square 
miles :  how  much  greater  is  the  Chinese  Empire  than  the 
United  States? 

21.  The  area  of  Europe  is  3781280  square  miles:  how 
much  greater  is  Euroj>e  than  the  United  States?  The  Chi¬ 
nese  Empire  than  Europe  ? 

22.  In  1866,  Ohio  produced  99766822  bushels  of  corn, 
and  Illinois  155844350  bushels :  how  many  bushels  did  Illi¬ 
nois  produce  more  than  Ohio? 

23.  A  man  bought  a  farm  for  $5867,  and  built  upon  it  a 
house  at  a  cost  of  $1850,  and  then  sold  the  farm  for  $7250: 
how  much  did  he  lose? 

24.  An  estate  of  $13450  was  divided  between  a  widow  and 
two  children;  the  widow’s  share  was  $6340,  the  son’s  $1560 
less  than  the  widow’s,  and  the  rest  fell  to  the  daughter: 
what  was  the  daughter’s  share? 

25.  A  man  deposited  in  a  bank  at  one  time  $850,  at  an- 


SUBTRACTION. 


19 


other,  $367,  and  at  another,  $670;  he  then  drew  out  $480, 
and  $375:  how  much  money  had  he  still  in  bank? 

26.  A  man  bought  a  farm  for  $6450,  giving  in  exchange 
a  house  worth  $4500,  a  note  for  $1150,  and  paying  the  dif¬ 
ference  in  money :  how  much  money  did  he  pay  ? 

27.  A  grain  dealer  bought  15640  bushels  of  wheat,  and 
sold  at  one  time  3465  bushels,  at  another,  4205,  and  at  an¬ 
other,  1080:  how  many  bushels  remained? 

28.  A  has  320  acres  of  land ;  B  has  65  acres  more  than 
A ;  C  has  124  acres  less  than  both  A  and  B ;  and  D  has  as 
many  acres  as  both  A  and  C  less  the  number  of  acres  owned 
by  B.  How  many  acres  have  B,  C,  and  D  respectively? 
How  many  have  all? 

29.  From  45003  plus  13478,  take  their  difference. 

DEFINITIONS  AND  PRINCIPLES. 

28.  Subtraction  is  the  process  of  finding  the  difference 
between  two  numbers. 

The  Difference  is  the  number  found  by  taking  one 
number  from  another. 

When  the  subtrahend  is  a  part  of  the  minuend  the  difference  is 
called  the  Remainder.  (Art.  41.) 

The  31inuend  is  the  greater  number. 

The  Subtrahend  is  the  number  subtracted. 

29.  The  Sign  of  Subtraction  is  a  short  horizontal 
line,  — .  It  is  called  minus  or  less .  Thus,  12  —  5  is  read 
12  minus  5  or  12  less  5. 

30.  Principles. — 1.  The  minuend,  subtrahend,  and  differ¬ 
ence  are  like  numbers. 

2.  The  minuend  is  the  sum  of  the  subtrahend  and  difference. 

3.  If  the  minuend  and  subtrahend  be  equally  increased,  the 
difference  will  not  be  changed. 

4.  The  adding  of  10  to  a  term  of  the  minuend  and  1  to  the 
next  higher  term  of  the  subtrahend,  increases  the  minuend  and 
subtrahend  equally. 


20 


COMPLETE  ARITHMETIC. 


SECTION  V. 

MULTIPLICATION. 


MENTAL  PROBLEMS. 

1.  There  are  24  hours  in  a  day :  how  many  hours  in  7 
days  ?  In  9  days  ?  11  days  ? 

2.  There  are  60  minutes  in  an  hour:  how  many  minutes 
in  8  hours?  In  12  hours?  15  hours? 

3.  If  a  man  earn  $63  a  month,  and  spend  $48,  how  much 
will  he  save  in  12  months  ? 

4.  If  12  men  can  do  a  piece  of  work  in  15  days,  how  long 
will  it  take  one  man  to  do  it  ? 

5.  If  35  bushels  of  oats  will  feed  8  horses  25  days,  how 
long  will  they  feed  one  horse  ? 

6.  Two  men  start  from  the  same  place  and  travel  in  op¬ 
posite  directions,  one  at  the  rate  of  28  miles  a  day,  and  the 
other  at  the  rate  of  32  miles  a  day:  how  far  will  they  be 
apart  at  the  end  of  five  days  ? 

7.  Two  men  are  450  miles  apart :  if  they  approach  each 
other,  one  traveling  30  miles  a  day  and  the  other  35  miles  a 
day,  how  far  apart  will  they  be  at  the  end  of  6  days  ? 

8.  A  cask  has  two  pipes,  one  discharging  into  it  90  gallons 
of  water  an  hour,  and  the  other  drawing  from  it  75  gallons 
an  hour:  how  many  gallons  of  water  will  there  be  in  the 
cask  at  the  end  of  12  hours? 

9.  A  had  $24,  B  four  times  as  much  as  A  less  $16,  and 
C  twice  as  much  as  A  and  B  together  plus  $17 :  how  much 
money  had  B  and  C  ? 

10.  A  farmer  sold  to  a  grocer  15  pounds  of  butter,  at  30 
cents  a  pound,  and  bought  8  pounds  of  sugar,  at  15  cents 
a  pound,  and  9  pounds  of  coffee,  at  20  cents  a  pound :  how 
much  was  still  due  him? 


MULTIPLICATION. 


21 


WHITTEN  PROBLEMS. 

11.  Multiply  624  by  45 ;  by  405 ;  by  4005. 

12.  Multiply  38400  by  27  ;  by  607 ;  by  6007. 

13.  Multiply  7863  by  69 ;  by  6900  ;  by  64000. 

14.  Multiply  48000  by  760 ;  by  7600000. 

15.  There  are  5280  feet  in  a  mile :  how  many  feet  in  608 
miles?  In  3300  miles? 

16.  The  earth  moves  1092  miles  in  a  minute :  how  far 
does  it  move  in  1440  minutes,  or  one  day? 

17.  A  square  mile  contains  640  acres,  and  the  state  of 
Ohio  contains,  in  round  numbers,  40000  square  miles :  how 
many  acres  in  the  state  ? 

18.  If  a  garrison  of  380  soldiers  consume  56  barrels  of 
flour  in  75  days,  how  many  soldiers  will  the  same  amount 
of  flour  supply  one  day? 

19.  A  man  bought  a  farm,  containing  472  acres,  at  $24 
an  acre,  and  after  investing  $3450  in  buildings,  he  sold  the 
farm, at  $33  an  acre :  did  he  gain  or  lose,  and  how  much? 


DEFINITIONS  AND  PRINCIPLES. 


31.  Multiplication  is  a  process  of  taking  one  num¬ 
ber  as  many  times  as  there  are  units  in  another. 

The  3£ultiplicand  is  the  number  taken  or  multiplied. 

The  Multiplier  is  the  number  denoting  how  many 
times  the  multiplicand  is  taken. 

The  Product  is  the  number  obtained  by  multiplying. 


The  multiplicand  and  multiplier  are  Factors  of  the  product,  and 
the  product  is  a  Multiple  of  each  of  its  factors. 

32.  The  Sign  of  Multiplication  is  an  oblique 

cross,  X-  It  is  read  multiplied  by. 

WLpn  p]  p  p-pH  W.wppn  Imx-mimbers,  it  shows  that  they  are  to  be 
multiplied  together;  and,  since  the  order  of  the  factors  does  not 
affect  the  product,  either  number  may  be  made  the  multiplier. 
The  multiplier  is  usually  written  after  the  sign;  when  it  is  written 
before  it,  the  sign  is  read  times. 

,tt  _ 


Vi 


{OS'*} 


y 


j 


22 


COMPLETE  ARITHMETIC. 


33.  The  product  may  be  obtained  by  adding  the  multipli¬ 
cand  to  itself  as  many  times  less  one  as  there  are  units  in 
the  multiplier.  Hence,  Multiplication  is  a  short  method  of 
finding  the  sum  of  several  equal  numbers. 

34.  Principles. — 1.  The  Multiplicand  may  be  either  con¬ 
crete  or  abstract. 

2.  The  multiplier  must  always  be  regarded  as  abstract. 

3.  The  product  and  multiplicand  are  like  numbers. 

4.  The  product  is  not  affected  by  changing  the  order  of  the 
factors.  Thus,  4  X  3  =  3  X  4. 

5.  The  multiplicand  equals  the  product  divided  by  the  multi- 
• 

6.  The  multiplier  equals  the  product  divided  by  the  multipli¬ 
cand. 

7.  The  division  of  either  the  multiplicand  or  the  multiplier  by 
any  number  divides  the  product  by  that  number. 


ABBREVIATED  PROCESSES. 


Case  I. 

The  Multiplier  lO,  lOO,  lOOO,  etc. 

1.  There  are  7  days  in  a  week:  how  many  days  in  10 
weeks?  In  100  weeks? 

2.  There  are  24  hours  in  a  day :  how  many  hours  in  10 
days?  100  days? 

3.  If  a  railway  train  run  30  miles  an  hour,  how  far  will 
it  run  in  10  hours?  1000  hours? 

4.  If  a  freight  car  will  carry  18  head  of  cattle,  how  many 
cattle  will  10  cars  carry?  100  cars?  1000  cars? 

5.  There  are  12  months  in  a  year :  how  many  months  in 
100  years?  1000  years? 


WRITTEN  PROBLEMS. 

6.  Multiply  648  by  100. 


Process  :  648  X  100  =  64800.  The  annexing  of  a  cipher  to  a  rmm- 
ber  removes  the  significant  figures  one  place  to  the  left,  and  hence 
increases  their  value  10  times ;  the  annexing  of  two  ciphers  removes 

yrtA  .  1  ^  .  /] 


A  1 

U  &fcs. 


r 


^  $  <  *'V'' 

f 

A 

/ 

•Wi c/\  i 


MULTIPLICATION. 


23 


the  significant  figures  two  places  to  the  left,  and  increases  their  value 
100  times.  Hence,  the  annexing  of  two  ciphers  to  648  multiplies  it 
by  100. 

7.  Multiply  456  by  10 ;  by  10000. 

8.  Multiply  3050  by  100 ;  100000. 

9.  Multiply  347000  by  1000;  by  1000000. 

10.  Multiply  889000  by  10000 ;  by  100. 

f  35.  Principle. — The  removal  of  a  figure  one  order  to  the 
left  multiplies  its  value  by  10  (Art.  21). 


36.  Rule. — To  multiply  by  10,  100,  1000,  etc.,  Annex  to 
the  multiplicand  as  many  ciphers  as  there  are  ciphers  in  the  mul¬ 
tiplier . 

Case  II. 


The  Multiplier  a  convenient  part  of*  lO,  lOO, 

lOOO,  etc. 

Note. — If  the  class  is  not  sufficiently  familiar  with  the  subject  of 
fractions,  this  case  may  be  omitted. 

11.  There  are  24  sheets  of  paper  in  a  quire:  how  many 
sheets  in  2^  quires?  In  3^  quires? 

12.  There  are  60  minutes  in  an  hour :  how  many  minutes 
in  3^  hours?  In  12^-  hours? 

13.  If  a  workman  earn  $40  a  month,  how  much  will  he 
earn  in  2^  months?  In  12 J  months? 

14.  At  36  cents  a  yard,  what  will  25  yards  of  cloth  cost  ? 
33^  yards? 

15.  At  24  cents  a  dozen,  what  will  12£  dozens  of  eggs 
cost?  16|  dozens? 


WRITTEN  PROBLEMS. 

16.  Multiply  459  by  33  J. 

Process.  Since  33£  is  I  of  100,  33^  times  459  =  ^  of  100 

3  )  45900  times  459  =  1  of  45900.  Or,  multiply  the  multi- 

15300  Prod.  plicand  by  100,  and  divide  the  product  by  3. 

17.  Multiply  486  by  3-J ;  by  33^-. 

18.  Multiply  1688  by  12  J;  by  25;  by  50. 


24 


COMPLETE  ARITHMETIC. 


19.  Multiply  40648  byl6|;  by  33*;  by  333*. 

20.  Multiply  3468  by  25 ;  by  125 ;  by  250. 

21.  Multiply  4086  by  16|  by  166|;  by  333-J. 

22.  Multiply  10366  by  50 ;  by  33* ;  by  66§ 

37.  Principle. — If  the  multiplier  he  multiplied  hy  a  given 
number ,  and  the  resulting  product  he  divided  hy  the  same  num¬ 
ber,  the  quotient  will  he  the  true  product. 

38.  Rule. — To  multiply  by  a  convenient  part  of  10,  100, 
1000,  etc.,  Multiply  hy  10,  100,  1000,  etc.,  and  divide  the 
product  by  the  number  of  times  the  multiplier  has  been  increased . 

Case  III. 

The  ^Multiplier  a  little  less  than  lO,  lOO,  lOOO,  etc. 

23.  Multiply  467  by  98. 

Process.  Since  98  —  100  —  2,  the  product  of  467  by  98  =  467 

46700  X  100  —  467  X  2,  or  46700  —  934.  In  multiplying 

934  by  ioo  the  multiplicand  is  taken  two  times  more  than 

45766,  Prod,  it  should  be. 


24.  Multiply  5672  by  99 ;  by  999. 

25.  Multiply  40863  by  97 ;  by  997. 

26.  Multiply  8679  by  998 ;  by  9998. 

27.  Multiply  618734  by  95;  by  99995. 

39.  Rule. — To  multiply  by  a  number  a  little  less  than  10, 
100,  1000,  etc.,  Multiply  hy  10,  100,  1000,  etc.,  and  subtract 
from  the  product  the  multiplicand  multiplied  hy  the  difference  be¬ 
tween  the  multiplier  and  10,  100,  1000,  etc.,  as  the  case  may  he. 

Case  IV. 


The  Multiplier  14,  15,  16,  etc.,  or  31,  51,  61,  etc. 


28.  Multiply  7856  by  14;  by  41. 

1st  Process. 

7856  X  14 
31424 


2d  Process. 

7856  X  41 
31424 


109984,  Product.  322096,  P'oduct. 

Note. — An  inspection  of  each  process  will  suggest  its  explanation. 
The  second  partial  product  need  not  be  written,  as  the  successive  terms 
can  be  added  mentally  to  the  proper  terms  of  the  first  partial  product. 


DIVISION. 


25 


29.  Multiply  38407  by  13 ;  by  15 ;  by  17. 

30.  Multiply  4960  by  16 ;  by  18 ;  by  19. 

31.  Multiply  360978  by  31 ;  by  51 ;  by  71. 

32.  Multiply  48706  by  61 ;  by  81 ;  by  91. 

33.  Multiply  34087  by  17 ;  by  71 ;  by  18. 


40.  Rules. — 1.  To  multiply  by  13,  14,  15,  etc.,  Multiply 
bv  the  units’  term,  and  add  the  successive  products  after  the  first , 
which  is  units ,  to  the  successive  terms  of  the  multiplicand. 

2.  To  multiply  by  31,  41,  51,  etc.,  Multiply  by  the  tens’  term , 
and  add  the  successive  products  to  the  successive  terms  of  the  mul¬ 
tiplicand  beginning  with  tens. 


SECTION  VI. 

DIVISION. 


MENTAL  PROBLEMS. 

1.  There  are  7  days  in  a  week:  how  many  weeks  in  63 
days?  98  days?  126  days? 

2.  There  are  eight  quarts  in  a  peck:  how  many  pecks 
in  72  quarts?  120  quarts?  144  quarts? 

3.  There  are  60  minutes  in  an  hour :  how  many  hours  in 
480  minutes?  720  minutes?  1440  minutes? 

4.  A  man  paid  $3600  for  a  farm,  paying  at  the  rate  of 
$40  an  acre :  how  many  acres  in  the  farm  ? 

5.  A  grocer  bought  12  barrels  of  flour  for  $90,  and  sold 
them  so  as  to  gain  $18:  how  much  did  he  receive  per 
barrel  ? 

6.  Two  men  are  120  miles  apart,  and  are  traveling  toward 
each  other,  one  at  the  rate  of  7  miles  an  hour,  and  the 
other  at  the  rate  of  8  miles  an  hour:  in  how  many  hours 
will  they  meet? 

7.  If  a  man  can  build  a  wall  in  84  days,  how  long  will 
it  take  7  men  to  build  it  ? 

C.  Ar. — 3. 


26 


COMPLETE  ARITHMETIC. 


8.  If  8  men  can  do  a  piece  of  work  in  15  days,  how  long 
will  it  take  12  men  to  do  it? 

9.  If  a  quantity  of  provisions  will  supply  a  ship’s  crew 
of  20  men  15  weeks,  how  large  a  crew  will  it  supply  25 
weeks  ? 

10.  If  a  man  can  do  a  piece  of  work  in  40  days,  by  work¬ 
ing  8  hours  a  day,  how  long  would  it  take  him  if  he  should 
work  10  hours  a  day? 

11.  A  man  earns  $16  while  a  boy  earns  $9 :  how  many 
dollars  will  the  man  earn  while  the  boy  is  earning  $72  ? 

12.  The  fore  wheels  of  a  carriage  are  each  9  feet  in  cir¬ 
cumference,  and  the  hind  wheels  are  each  12  feet:  if  the 
fore  wheels  each  rotate  400  times  in  going  a  certain  distance, 
how  many  rotations  will  each  hind  wheel  make  ? 

13.  Five  times  Harry’s  age  plus  4  times  his  age,  minus  6 
times  his  age,  plus  7  times  his  age,  minus  5  times  his  age, 
equals  60  years :  how  old  is  Harry  ? 

14.  A  number  multiplied  by  6,  divided  by  3,  multi¬ 
plied  by  8,  and  divided  by  4,  equals  96 :  what  is  the  num¬ 
ber  ? 

WRITTEN  PROBLEMS. 

15.  Divide  486  by  6 ;  by  8 ;  by  9. 

16.  Divide  8408  by  12;  by  24;  by  36. 

17.  Divide  84600  by  900;  by  12000. 

18.  Divide  412304  by  3600;  by  303000. 

19.  The  dividend  is  1059984  and  the  divisor  is  306 :  what 
is  the  quotient? 

20.  The  dividend  is  2185750  and  the  quotient  is  250:  what 
is  the  divisor? 

21.  The  product  is  1123482  and  the  multiplier  is  246: 
what  is  the  multiplicand? 

22.  How  many  passenger  cars,  costing  $2450  each,  can  be 
bought  for  $100450? 

23.  There  are  5280  feet  in  a  mile,  and  the  height  of 
Mount  Everest,  in  Asia,  is  29100  feet :  what  is  its  height  in 
miles  ? 


DIVISION. 


27 


24  There  are  3600  seconds  in  an  hour :  how  many  hours 
in  738000  seconds  ? 

DEFINITIONS  AND  PRINCIPLES. 

41.  Division  is  the  process  of  finding  how  many 
times  one  number  is  contained  in  another;  or,  it  is  the 
process  of  finding  one  of  the  equal  parts  of  a  number. 

The  Dividend  is  the  number  divided. 

The  Divisor  is  the  number  by  which  the  dividend  is 
divided. 

The  Quotient  is  the  number  of  times  the  divisor  is  con¬ 
tained  in  the  dividend ;  or  it  is  one  of  the  equal  parts  of  the 
dividend. 

The  j Remainder  is  the  part  of  the  dividend  which  is 
left  undivided. 

42.  The  Sign  of  Division  is  a  short  horizontal 
line  between  two  dots,  -f-.  It  is  read  divided  by.  Thus, 
16  -i-  4  is  read  16  divided  by  4. 

Division  is  also  expressed  by  writing  the  dividend  above  and  the 

divisor  below  a  short  horizontal  line.  Thus,  ^  is  rea(i  18  divided  by  3. 

/ 

43.  There  are  two  methods  of  division,  called  Short  Di¬ 
vision  and  Long  Division. 

In  Short  Division,  the  partial  products  and  partial 
dividends  are  not  written,  but  are  formed  mentally. 

In  Long  Division,  the  partial  products  and  partial 
dividends  are  written. 

44.  — 1.  One  number  is  contained  in  another  as  many 
times  as  it  must  be  taken  to  produce  it.  Hence,  Division  is 
the  reverse  of  multiplication. 

2.  One  number  is  contained  in  another  as  many  times  as 
it  can  be  taken  from  it.  Hence,  Division  is  a  brief  method 
of  finding  how  many  times  one  number  jian  be  subtracted  from 
another. 


28 


COMPLETE  ARITHMETIC. 


45.  Principles. — 1.  The  divisor  and  quotient  are  factors  of 

the  dividend. 

2.  When  division  finds  how  many  times  one  number 
is  contained  in  another,  the  divisor  and  dividend  are  like 
numbers,  and  the  quotient  is  an  abstract  number. 

3.  When  division  finds  one  of  the  equal  parts  of  a  num¬ 
ber,  the  divisor  is  an  abstract  number,  and  the  dividend  and  quo¬ 
tient  are  like  numbers. 

4.  The  multiplying  of  both  divisor  and  dividend  by  the  same 
number  does  not  change  the  quotient. 

5.  The  dividing  of  both  dividend  and  divisor  by  the  same  num¬ 
ber  does  not  change  the  quotient. 


ABBREVIATED  PROCESSES. 

Case  I. 

The  Divisor  lO,  lOO,  lOOO,  etc. 

1.  There  are  10  cents  in  a  dime:  how  many  dimes  in  80 
cents?  120  cents?  240  cents? 

2.  There  are  10  dimes  in  a  dollar:  how  many  dollars  in 
70  dimes?  250  dimes?  2500  dimes? 

3.  There  are  100  cents  in  a  dollar :  how  many  dollars  in 
800  cents?  2400  cents?  7500  cents? 

4.  At  $10  a  barrel,  how  many  barrels  of  flour  can  be 
bought  for  $90?  For  $150? 

5.  At  $100  apiece,  how  many  horses  can  be  bought  for 
$1200  ?  For  $2500  ?  For  $45000  ? 


WRITTEN  problems. 


6.  Divide  450  by  10. 

Process. 

4510 

45,  Quotient. 


7.  Divide  3852  by  100. 

Process. 

38152 

38,  Quotient. 

52,  Remainder. 


The  explanation  of  these  processes  is  obvious.  The  cutting  off  of 
the  right-hand  figure  removes  all  the  other  figures  one  place  to  the 
right,  and  thus  decreases  their  value  ten  times.  The  cutting  off  of 
two  figures  removes  the  other  figures  two  places  to  the  right,  and  de- 


DIVISION. 


29 


creases  their  value  one  hundred  times.  The  figures  cut  off  denote  the 
remainder. 

8.  Divide  356000  by  100 ;  by  1000. 

9.  Divide  46035  by  100 ;  by  1000. 

10.  Divide  384602  ;  by  1000 ;  by  10000. 

11.  Divide  95000000  by  10000;  by  1000000. 

46.  Principle. — The  removal  of  a  figure  one  order  to  the 
right  divides  its  value  by  10  (Art.  21). 

47.  Rule. — To  divide  by  10,  100,  1000,  etc.,  Cut  off  as 
many  figures  from  the  right  of  the  dividend  as  there  are  ciphers 
in  the  divisor.  The  figures  cut  off  denote  the  remainder. 

Case  II. 

Tine  Divisor  ending  with,  one  or  more  Ciphers. 

12.  There  are  20  quires  of  paper  in  a  ream :  how  many 
reams  in  80  quires?  160  quires? 

13.  There  are  fifty  cents  in  a  half-dollar :  how  many  half- 
dollars  in  150  cents  ?  350  cents  ? 

14.  There  are  60  minutes  in  an  hour :  how  many  hours 
in  240  minutes?  720  minutes? 

15.  A  barrel  of  beef  contains  200  pounds :  how  many 
barrels  will  1200  pounds  make?  3600  pounds? 


WRITTEN  PROBLEMS. 


16.  Divide  71400  by  3400. 


Process. 

34|00)714|00(21 

68 

34 

34 


First  divide  both  divisor  and  dividend  by 
100,  which  is  done  by  cutting  off  the  two 
right-hand  figures.  Then  divide  714,  the 
new  dividend,  by  34,  the  new  divisor. 


17.  Divide  58864  by  4500. 


Process. 

45|00)  588I64(13 
45 

138 

135 

3 

Remainder ,  364 


First  divide  both  dividend  and  divisor  by 
100,  which,  in  the  case  of  the  dividend,  leaves 
a  remainder  of  64.  Next  divide  588  by  45, 
leaving  a  remainder  of  3,  which  is  3  hundreds 
since  the  dividend  (588)  is  hundreds.  The  first 
remainder  is  64  units  which,  annexed  to  the 
3  hundreds,  give  364,  the  true  remainder. 


30 


COMPLETE  ARITHMETIC. 


18.  Divide  63200  by  7900  ;  by  7000. 

19.  Divide  116000  by  2500;  by  4800. 

20.  Divide  172800  by  14400;  by  18000. 

21.  Divide  129600  by  4800;  by  64000. 

48.  Principle. — The  dividing  of  both  divisor  and  dividend 
by  the  same  number  does  not  change  the  quotient. 

49.  Rule. — To  divide  by  a  number  ending  in  one  or  more 
ciphers,  1.  Cut  off  the  ciphers  from  the  right  of  the  divisor , 
and  an  equal  number  of  figures  from  the  right  of  the  dividend. 

2.  Divide  the  new  dividend  thus  formed  by  the  new  divisor, 
and  the  result  will  be  the  quotient. 

3.  Annex  the  figures  cut  off  from  the  dividend  to  the  remainder, 
if  there  be  one,  and  the  result  will  express  the  true  remainder. 


Case  III. 

The  Divisor  a.  convenient  part  of  lO,  lOO,  etc. 

22.  At  3J  cents  apiece,  how  many  lemons  can  be  bought 
for  90  cents?  For  240  cents? 

Suggestion. — Since  10  is  3  times  3£,  multiply  the  dividend  by  3 
and  divide  the  product  by  10. 

23.  At  12^-  cents  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  75  cents?  For  225  cents? 

24.  At  16|  cents  a  bushel,  how  many  bushels  of  coal  can 
be  bought  for  150  cents?  For  550  cents? 

25.  At  $33|  a  head,  how  many  cows  can  be  bought  for 
$200  ?  For  $1200  ? 


WRITTEN  EXERCISES. 


26.  Divide  4375  by  125. 

Process. 

4375 

8 

351000 

35,  Quotient. 


27.  Divide  13600  by  333£. 

Process. 

13600 

3 

401800 

40,  Quotient. 

800 -f-3  =  2 66§  Bern. 


28.  Divide  6250  by  33 J;  by  50. 


DIVISION. 


31 


29.  Divide  4365  by  250 ;  by  166§. 

30.  Divide  15300  by  16f ;  by  3331. 

50.  Principle. — The  multiplying  of  both  divisor  and  divi¬ 
dend  by  the  same  number  does  not  change  the  quotient . 

51.  Rule. — To  divide  by  a  convenient  part  of  10,  100, 
1000,  etc.,  Multiply  the  divideTid  by  the  number  denoting  how 
many  times  the  divisor  is  contained  in  10,  or  100,  or  1000,  etc., 
and  divide  the  product  by  10,  or  100,  or  1000,  etc. 


Case  IV. 

The  Divisor  a  Composite  Number. 


31.  Divide  18315  by  45. 


Process. 


Illustrative  Process. 


45  =  5  X  9 
5 ) 18315 
9)3663 

407,  Quotient. 


5  )  18315  -4-  45  =  3663  -4-  9 
9)3663-=-  9=  407-4-1 
407-4-  1=  407 


Since  45=  5  X  9,  the  quotient  obtained  by  dividing  18315  by  5,  is 
9  times  too  large ,  and  hence  this  quotient  (3663)  divided  by  9,  is  the 
true  quotient. 

The  process  of  dividing  by  the  factors  of  the  divisor  successively 
is  the  same  in  principle  as  the  division  of  both  dividend  and  divisor 
by  these  factors  successively,  which  (Art.  48)  does  not  change  the 
value  of  the  quotient.  See  “  Illustrative  Process.” 


32.  Divide  58636  by  28 ;  by  77. 

33.  Divide  13328  by  49  ;  by  56  ;  by  70. 

34.  Divide  31360  by  64;  by  70;  by  81. 

35.  Divide  3687  by  64. 

Process. 

2  )_3687  64  =  2  X  8  X  4 

8)1843....  1  (1st  Rem.)  = . 1 

4  )  230  ....  3  (2d  “  )=3X2=  ....  6 

57  ....  2  (3d  “  )  =  2  X  8  X  2  =  32 

True  Remainder  =  39 


A  unit  of  the  first  quotient  equals  2  units  of  the  dividend,  and 
hence  the  second  remainder  (3)  equals  3X2  units  of  the  dividend. 


32 


COMPLETE  ARITHMETIC. 


A  unit  of  the  second  quotient  equals  8  units  of  the  first  quotient,  and 
hence  the  third  remainder  (2)  equals  2X8  units  of  the  second  quo¬ 
tient  =  2X8X2  units  of  the  dividend.  Hence  the  first  remainder 
is  1 ;  the  second  6 ;  the  third  32 ;  and  the  total,  or  true  remainder,  39. 

Note. — The  teacher  can  illustrate  this  process  by  considering  the 
dividend  (3687)  'pints.  The  first  quotient  will  be  quarts,  the  second 
pecks,  and  third  bushels,  and  the  first  remainders  will  be  1  pt.,  the 
second,  3  qt.,  and  the  third,  2  pk.  1  pt.  -f  3  qt.  -f-  2  pk.  =  39  pt. 

36.  Divide  34567  by  63 ;  by  72. 

37.  Divide  120473  by  56 ;  by  81. 

38.  Divide  400671  by  64;  by  77. 

39.  Divide  346000  by  55 ;  by  96. 

40.  Divide  47633  by  90;  by  110. 

52.  Principle. — The  division  of  both  divisor  and  dividend 
by  the  same  number  does  not  change  the  quotient. 

53.  Rule. — To  divide  by  a  composite  number,  1.  Resolve 
the  divisor  into  convenient  factors ;  divide  the  dividend  by  one  of 
these  factors,  the  quotient  thus  obtained  by  another,  and  so  on 
until  all  the  factors  are  used  as  divisors.  The  last  quotient  will 
be  the  true  quotient. 

2.  Multiply  each  remainder,  except  the  first,  by  all  the  divisors 
preceding  its  own.  The  sum  of  these  products  and  the  first  re¬ 
mainder  will  be  the  true  remainder. 


SECTION  VII. 

/ 

PROPERTIES  OF  NUMBERS. 

DIVISORS  AND  FACTORS. 

Note. — The  terms  number,  divisor,  and  factor,  used  in  this  section, 
denote  integral  numbers. 

1.  What  two  numbers  besides  itself  and  1  will  exactly 
divide  10?  21?  35?  63?  77? 

2.  What  numbers  besides  itself  and  1  will  exactly  divide 
7?  11?  17?  23?  37?  41? 


PROPERTIES  OF  NUMBERS. 


33 


3.  What  numbers  will  exactly  divide  15?  13?  28?  29? 
42?  43? 

Note. — Since  every  integer  is  exactly  divisible  by  itself  and  1, 
these  divisors  need  not  be  given. 

4.  What  numbers  will  exactly  divide  30  ?  31  ?  45  ?  53  ? 
56?  67?  65? 

5.  Name  all  the  prime  numbers  between  0  and  20;  30 
and  50. 

6.  Name  all  the  composite  numbers  between  20  and  30; 
50  and  70. 

7.  What  are  the  prime  divisors  of  6?  15?  18?  21?  30? 
45?  50?  54? 

8.  What  are  the  prime  factors  of  12  ?  24  ?  35  ?  39  ?  42  ? 

9.  What  are  the  prime  factors  of  27  ?  36  ?  49  ?  56  ?  63  ? 
66?  72?  84? 

10.  Of  what  numbers  are  2  and  5  prime  factors?  2,  3, 
and  5  ?  2,  5,  and  7  ?  3,  5,  and  7  ? 

11.  Of  what  numbers  are  2,  2,  and  3  prime  factors?  2,  3, 
3,  and  5  ?  2,  3,  5,  and  7  ? 

12.  What  prime  factor  is  common  to  9  and  12?  15  and 
25?  18  and  30?  21  and  28? 

13.  What  prime  factor  is  common  to  24  and  27  ?  35  and 
42  ?  44  and  77  ?  35  and  50  ?  63  and  70  ? 


WRITTEN  EXERCISES. 


14.  What  are  the  prime  factors  of  126  ? 


Process. 

2)126 

3)_63 

3)_21 

7 

126  =  2  X  3  X  3  X  7. 


Divide  126  by  2,  a  prime  divisor ;  next 
divide  the  quotient  63  by  3,  a  prime 
divisor,  and  then  divide  the  quotient  21 
by  3,  a  prime  divisor.  The  prime  factors 
are  2,  3,  3,  and  7. 


What  are  the  prime  factors  of 

15.  160?  18.  325?  21.  462?  24.  748? 

16.  175?  19.  330?  22.  490?  25.  693? 

17.  256?  20.  420?  23.  594?  26.  1155? 


34 


COMPLETE  ARITHMETIC. 


What  prime  factors  are  common  to 

27.  45  and  63  ?  30.  200  and  250  ? 

28.  50  and  80?  31.  175  and  325? 

29.  96  and  256?  32.  144  and  180? 

DEFINITIONS,  PRINCIPLES,  AND  RULES. 

54.  The  j Divisor  of  a  number  is  any  number  that  will 
exactly  divide  it. 

55.  Numbers  are  either  Prime  or  Composite . 

A  Prime  Number  has  no  divisor  except  itself  and 
one. 

A  Composite  Number  has  other  divisors  besides 
itself  and  one. 

Every  composite  number  is  the  product  of  two  or  more  numbers. 

56.  Two  or  more  numbers  are  prime  to  each  other,  or  rela¬ 
tively  prime,  when  they  have  no  common  divisor  except  1. 
Thus,  9  and  16  are  prime  to  each  other. 

All  prime  numbers  are  prime  to  each  other.  Composite  numbers 
may  be  relatively  prime,  as  9  and  10;  16  and  25. 

57.  A  Factor  of  a  number  is  its  divisor. 

A  Prime  Factor  of  a  number  is  its  prime  divisor. 

The  terms  divisor  and  factor  differ  only  in  their  use,  the  former 
implying  division  and  the  latter  multiplication.  A  divisor  or  factor 
of  a  number  is  also  called  its  measure. 

58.  When  a  number  is  a  factor  of  each  of  two  or  more 
numbers,  it  is  called  their  Common  Factor,  Thus,  5 
is  a  common  factor  of  15  and  20. 

59.  Whole  numbers  are  either  Even  or  Odd. 

An  j Even  Number  is  exactly  divisible  by  2 ;  as,  2,  4, 
6,  8,  10,  12,  etc. 

An  Odd  Number  is  not  exactly  divisible  by  2 ;  as,  1, 
3,  5,  7,  9,  11,  13,  etc. 


CANCELLATION. 


35 


All  the  even  numbers  except  2  are  composite.  Some  of  the  odd 
numbers  are  composite  and  others  are  prime. 

60.  Principles. — 1.  A  factor  of  a  number  is  a  factor  of 
any  number  of  times  that  number. 

2.  A  common  factor  of  two  or  more  numbers  is  a  factor  of 
their  sum. 

3.  A  composite  number  is  the  product  of  all  its  prime  factors. 

A.  If  a  composite  number  composed  of  two  factors  be  divided 

by  one  factor ,  the  quotient  will  be  the  other  factor. 

5.  If  any  composite  number  be  divided  by  a  factor ,  or  by  the 
product  of  any  number  of  its  factors,  the  quotient  will  be  the 
product  of  the  remaining  factors. 

61.  Rules. — 1.  To  resolve  a  composite  number  into  its 
prime  factors,  Divide  it  by  any  prime  divisor,  and  the  quo¬ 
tient  by  any  prime  divisor,  and  so  continue  until  a  quotient  is 
obtained  which  is  a  prime  number.  The  several  divisors  and 
the  last  quotient  are  the  priine  factors. 

2.  To  find  the  common  factors  of  two  or  more  numbers, 
Resolve  the  given  numbers  into  their  prime  factors  and  select  the 
factors  which  are  found  in  all  the  numbers 

\  CANCELLATION. 

33.  Divide  the  product  of  4,  7,  9,  and  12  by  the  product 
of  4,  7,  and  9. 


Process. 

Dividend,  4  X  t  X  0  X  12 
Divisor,  4  X  #  X  0 


=  12. 


Instead  of  forming  the  prod¬ 
ucts,  indicate  the  multiplica¬ 
tion  by  the  proper  sign,  and 
write  the  divisor  underneath 


the  dividend.  Since  dividing  both  dividend  and  divisor  by  the  same 
number  does  not  affect  the  value  of  the  quotient  (Art.  48),  divide 
each  by  4,  7,  and  9.  This  may  be  done  by  canceling,  as  indicated  in 
the  process.  The  quotient  is  12. 


34.  Multiply  4  X  7  by  12,  and  divide  the  product  bj  4 
times  12. 

35.  Divide  6  X  8  X  20  by  4  X  20. 

36.  Divide  5  X  7  X  11  X  13J  by  7  X  13^. 


36 


COMPLETE  ARITHMETIC. 


37.  Divide  12  X  16  X  28  by  9  X  24  X  21. 


Process. 

8  4 


Since  dividing  the  factor 
of  a  number  divides  the  num- 


1%  X  X  8X4  32  her,  cancel  12  in  the  divi- 

9  x  H  X  %l  ~  9  X3=T7  =  1*y  dend  and  divide  24  in  the 
%  3  divisor  by  12,  giving  2.  Can¬ 

cel  the  2  and  divide  16  in  the 
dividend  by  2,  giving  8.  Divide  the  28  in  the  dividend  and  21  in 
the  divisor,  each  by  7,  giving  4  and  3.  The  uncanceled  factors  of 
the  divisor  are  8  and  4,  and  those  of  the  dividend  are  9  and  3.  The 
quotient  is  32  -f-  27  —  l/7. 


38.  Divide  24  X  27  X  12^-  by  18  X  54  X  50. 

39.  Divide  28  X  30  X  100  by  21  X  15  X  33|. 

40.  40  X  22  X  35  X  16f -f- (20  X  44  X  50  X  49)  =  wbat? 

41.  A  farmer  exchanged  12  barrels  of  apples,  each  con¬ 
taining  3  bushels,  at  75  cts.  a  bushel,  for  25  sacks  of  pota¬ 
toes,  each  containing  2  bushels :  how  much  did  the  potatoes 
cost  a  bushel? 

42.  If  9  men  can  do  a  piece  of  work  in  16  days,  working 
10  hours  a  day,  how  many  men  can  do  it  in  20  days,  work¬ 
ing  8  hours  a  day  ? 


DEFINITION,  PRINCIPLES,  AND  RULE. 

62.  Cancellation  is  the  omission  of  one  or  more  of 
the  equal  factors  of  divisor  and  dividend.  It  is  used  to 
abbreviate  the  process  of  division. 

63.  Principles. — 1.  The  canceling  of  one  of  the  factors  of 
a  number  divides  the  number  by  the  factor  canceled. 

2.  Canceling  equal  factors  of  both  dividend  and  divisor 
divides  them  by  the  same  number ,  and  hence  does  not  change  the 
quotient. 

3.  Dividing  one  of  the  composite  factors  of  a  product  divides 
the  product. 

64.  Rule. — Indicate  the  multiplications  by  the  proper  sign , 
and  write  the  divisor  underneath  the  dividend.  Cancel  the  fac- 


COMMON  DIVISOR. 


37 


tors  common  to  both  dividend  and  divisor,  and  divide  the  prod¬ 
uct  of  the  factors  left  in  the  dividend  by  the  product  of  those 
left  in  the  divisor. 

Note. — When  all  the  expressed  factors  of  either  dividend  or  divisor 
are  canceled,  1  remains  as  a  factor. 


GREATEST  COMMON  DIVISOR 

1.  What  are  the  divisors  of  15?  28?  45?  53?  75?  90? 
91?  108? 

2.  What  is  a  common  divisor  of  15  and  35?  42  and  56? 
63  and  72  ?  64  and  80  ? 

3.  What  is  a  common  divisor  of  27  and  36  ?  18,  30,  and 
42  ?  36,  54,  and  72  ? 

4.  What  is  the  greatest  number  that  will  exactly  divide 
32  and  48?  45  and  90?  60  and  96? 

5.  What  is  the  greatest  common  divisor  of  36  and  60? 
45,  60,  and  75  ?  18,  54,  and  90  ? 

6.  What  is  the  greatest  common  divisor  of  24,  48,  and 
72?  16,  48,  and  80?.  20,  31,  and  45? 

7.  Show  that  every  common  divisor  of  12  and  16  is  a 
divisor  of  28,  their  sum. 

8.  Show  that  a  common  divisor  of  any  two  numbers  is  a 
divisor  of  their  sum. 

9.  Show  that  every  common  divisor  of  16  and  28  is  a 
divisor  of  12,  their  difference. 

10.  Show  that  a  common  divisor  of  any  two  numbers  is  a 
divisor  of  their  difference. 


WRITTEN  EXERCISES. 

11.  What  is  the  greatest  common  divisor  of  126  and  210? 


,  Process  by  Factoring. 
126  =$X$X3X# 

210  =  gX$X5Xtf 


2X3X7  =  42,  G.C.D. 
to  126  and  210  will  be  their  greatest  common  divisor. 


Resolve  126  and  210  into  their  prime 
factors.  Since  every  divisor  of  a  num¬ 
ber  is  a  prime  factor,  or  the  product  of 
two  or  more  prime  factors,  the  prod¬ 
uct  of  all  the  prime  factors  common 


38 


COMPLETE  ARITHMETIC. 


What  is  the  greatest  common  divisor  of 

12.  60  and  84?  15.  112,  140,  and  168? 

13.  63  and  126?  16.  84,  126,  and  210? 

14.  144  and  192?  17.  128,  256,  and  1280? 

18.  What  is  the  greatest  common  divisor  of  288  and  528  ? 


Divide  528  by  288,  and  288  by  the 
first  remainder  240,  and  240  by  the  sec¬ 
ond  remainder  48;  and,  there  being 
no  remainder,  48  is  the  greatest  com¬ 
mon  divisor  of  288  and  528. 

Since  48,  the  greatest  divisor  of  itself, 
is  a  divisor  of  240,  it  is  the  G.  C.  D. 
of  48  and  240. 


Process  by  Dividing. 

288  )  528  ( 1 
288 

240 )  288  ( 1 
240 

48  )  240  ( 5 
240 

48=  G.C.D.oi  288  and  528. 

Since  48  is  a  common  divisor  of  48  and  240,  it  is  a  divisor  of 
288,  their  sum ;  and  since  every  common  divisor  of  240  and  288  is 
a  divisor  of  48,  their  difference,  48,  the  greatest  divisor  of  itself,  is 
the  G.  C.  D.  of  240  and  288. 

Since  48  is  a  common  divisor  of  240  and  288,  it  is  a  divisor  of 
528,  their  mm;  and  since  every  common  divisor  of  288  and  528  is 
a  divisor  of  240,  their  difference,  48,  the  gi'eatest  common  divisor  of 
240  and  288,  is  the  G.  C.  D.  of  288  and  528. 


Note. — Let  the  pupil  show,  in  like  manner,  that  the  last  divisor, 
in  the  solution  of  problems  19  and  20,  is  the  greatest  common 
divisor  required. 


What  is  the  greatest  common  divisor  of 


19.  196  and  1728? 

20.  336  and  576  ? 

21.  407  and  888? 

22.  326  and  807  ? 

23.  756  and  1764? 

24.  1064  and  1274? 

25.  768  and  5184? 

26.  741  and  1938? 


27.  $260  and  $416? 

28.  $1815  and  $3465? 

29.  21451b.  and  34711b.? 

30.  175,  225,  and  275? 

31.  240,  360,  and  480? 

32.  144,  216,  and  648? 

33.  140,  308,  and  819  ? 

34.  240,  336,  and  1768? 


35.  What  is  the  greatest  common  divisor  of  1065,  1730, 
and  2845? 


36.  What  is  the  greatest  common  divisor  of  156,  585, 
442,  and  1287? 

37.  What  is  the  greatest  common  divisor  of  2731  and  3120? 


LEAST  COMMON  MULTIPLE. 


39 


DEFINITIONS,  PRINCIPLES,  AND  RULES. 

65.  A  Divisor  of  a  number  is  a  number  that  will  ex¬ 
actly  divide  it. 

A  Common  Divisor  of  two  or  more  numbers  is  a 
number  that  will  exactly  divide  each  of  them. 

The  Greatest  Common  Divisor  of  two  or  more 
numbers  is  the  greatest  number  that  will  exactly  divide  each 
of  them. 

66.  Principles. — 1.  Every  prime  factor,  and  every  product 
of  any  two  or  more  prime  factors  of  a  number,  is  a  divisor  of 
that  number.  Conversely, 

2.  Every  divisor  of  a  number  is  a  prime  factor,  or  the  product 
of  two  or  more  of  its  prime  factors. 

3.  The  product  of  all  the  prime  factors  common  to  two  or  more 
numbers  is  their  greatest  common  divisor. 

4.  The  divisor  of  a  number  is  a  divisor  of  any  number  of 
times  that  number. 

5.  A  common  divisor  of  two  numbers  is  a  divisor  of  their 
sum,  or  of  their  difference. 

6.  Any  common  divisor  of  either  of  two  numbers  and  their 
difference  is  a  common  divisor  of  the  two  numbers. 

67.  Rules. — 1.  To  find  the  greatest  common  divisor  of  two 
or  more  numbers  by  factoring,  Resolve  the  given  numbers  into 
their  prime  factors,  and  select  the  factors  which  are  common. 
The  product  of  the  common  factors  will  be  the  greatest  common 
divisor. 

2.  To  find  the  greatest  common  divisor  of  two  numbers 
by  division,  Divide  the  greater  number  by  the  less,  and  the 
divisor  by  the  remainder,  and  the  second  divisor  by  the  second  re¬ 
mainder,  and  so  on  until  there  is  no  remainder.  The  last  divisor 
will  be  the  greatest  common  divisor. 

Note. — When  there  are  three  or  more  numbers,  first  find  the  great¬ 
est  common  divisor  of  two  of  them,  and  then  the  greatest  common 
divisor  of  this  G.  C-  D.  and  a  third  number,  and  so  on. 


40 


COMPLETE  ARITHMETIC. 


LEAST  COMMON  MULTIPLE. 

1.  What  number  will  16  exactly  divide?  25?  30?  45? 

Note. — A  number  will  exactly  divide  its  multiple. 

2.  What  number  is  a  multiple  of  15?  24?  32?  54?  75? 
100?  120?  150?  200? 

3.  How  many  multiples  has  every  number? 

4.  What  number  will  8  and  10  both  exactly  divide?  9 
and  12?  20  and  25? 

5.  What  number  is  a  common  multiple  of  5  and  12?  15 
and  30?  25  and  50? 

6.  How  many  common  multiples  have  two  or  more  num¬ 
bers  ? 

7.  What  is  the  least  number  that  7  and  8  will  both  ex¬ 
actly  divide?  9  and  12?  20  and  30?  25  and  75? 

8.  What  number  is  the  least  common  multiple  of  7  and 
10?  12  and  18?  8,  12,  and  16? 

9.  How  many  least  common  multiples  have  two  or  more 
numbers  ? 

10.  Show  that  all  the  prime  factors  of  a  number  are 
factors  of  its  multiple,  and,  conversely,  that  a  number  con¬ 
taining  all  the  prime  factors  of  another  number  is  its  mul¬ 
tiple. 


WRITTEN  EXERCISES. 


11.  What  is  the  least  common  multiple  of  12,  18,  and  30? 


Process  by  Factoring. 

12  =  $  X  $  X  3 
18  =  2  X  $  X  $ 

80  =  2X  3  X  $ 

2X2X3X3X5  =  180,  L.  C.  M. 


Resolve  the  numbers  into 
their  prime  factors,  and  select 
all  the  different  factors,  re¬ 
peating  each  as  many  times  as 
it  is  found  in  any  number. 
The  factor  2  is  found  twice  in 
12;  the  factor  3,  twice  in  18; 
and  the  factor  5,  once  in  30.  The  product  of  2  X  2  X  3  X  3  X  5  is 
the  least  common  multiple  required,  since  it  is  the  least  number  which 
contains  all  the  prime  factors  of  12,  18,  and  30. 


LEAST  COMMON  MULTIPLE. 


41 


What  is  the  least  common  multiple  of 


12. 

8, 

12, 

20? 

16. 

18, 

24, 

72, 

48? 

13. 

9, 

21, 

42? 

17. 

15, 

35, 

70, 

105? 

14. 

32, 

48, 

80? 

18. 

25, 

75, 

100, 

150? 

15. 

27, 

54, 

108? 

19. 

$16, 

$40, 

$60, 

$72? 

20.  What  is  the  least  common  multiple  of  12,  15,  42,  70  ? 


Find  all  the  prime  factors 
by  dividing  the  given  num¬ 
bers  by  any  prime  number 
that  will  exactly  divide  two  or 
more  of  them,  thus :  Dividing 
by  2,  it  is  found  to  be  a  prime 
factor  of  12, 42,  and  70.  Write 
2X3X5X7X2  =  420,  L.  C.  M.  the  quotients  with  the  15  un¬ 
derneath.  Dividing  by  3,  it 
is  found  to  be  a  prime  factor  of  6,  15,  and  21,  and  hence  it  is  a  prime 
factor  of  12,  15,  and  42.  Dividing  by  5,  it  is  found  to  be  a  prime 
factor  of  5  and  35,  and  hence  of  15  and  70.  Dividing  by  7,  it  is 
found  to  be  a  prime  factor  of  7  and  7,  and  hence  of  42  and  70.  The 
remaining  quotient  2  is  a  prime  factor  of  12. 

Hence,  all  the  prime  factors  of  12,  15,  42,  and  70  are  2,  3,  5,  7,  and 
2,  and  since  the  product  of  these  several  prime  factors  (2  X  3  X  5  X 
7  X  2  =  420)  is  the  least  number  that  contains  each  of  them,  it  is  the 
least  common  multiple  of  12,  15,  42,  and  70. 


Process  by  Division. 


! )  12 

15 

42 

70 

3)6 

15 

21 

35 

5)2 

5 

7 

35 

7)2 

1 

7 

7 

2 

1 

1 

1 

What  is  the  least  common  multiple  of 


21.  12,  18,  30? 


26.  30,  45,  48,  80,  120? 


22.  8,  28,  70  ? 

23.  9,  20,  15,  36? 

24.  15,  24,  25,  30? 

25.  18,  21,  27,  36? 


27.  16,  30,  40,  50,  75  ? 

28.  15,  27,  35,  42,  70? 

29.  8,  28,  20,  24,  32,  48? 

30.  2,  3,  4,  5,  6,  7,  8,  9? 


DEFINITIONS,  PRINCIPLES,  AND  RULES. 

68.  A  Multiple  of  a  number  is  any  number  which  it 
will  exactly  divide. 

Note. — Every  number  is  an  exact  divisor  of  its  multiple. 


C.  Ar.— 1. 


42 


COMPLETE  ARITHMETIC. 


A  CoiriWlOTl  Multiple  of  two  or  more  numbers  is 
any  number  which  each  of  them  will  exactly  divide. 

The  Least  (Jotyityiou  Multiple  of  two  or  more 
numbers  is  the  least  number  which  each  of  them  will  exactly 
divide. 

Note. — The  following  definitions  may  be  preferred : 

A  Multiple  of  a  number  is  the  product  arising  from  taking  it 
two  or  more  times.  Or, 

A  Multiple  of  a  number  is  any  number  of  which  it  is  a  factor. 

A  Common  Multiple  of  two  or  more  numbers  is  a  multiple  of 
each  of  them. 

The  Least  Common  Multiple  of  two  or  more  numbers  is  the 
least  multiple  of  each  of  them. 

69.  Principles. — 1.  Every  multiple  of  a  number  contains 
all  its  prime  factors. 

2.  A  common  multiple  of  two  or  more  numbers  contains  all 
their  prime  factors. 

3.  The  least  common  multiple  of  two  or  more  numbers  con¬ 
tains  all  their  prime  factors,  and  no  other  factors. 

4.  The  least  common  multiple  of  two  or  more  numbers  con¬ 
tains  each  of  their  prime  factors  the  greatest  number  of  times  it 
occurs  in  either  number. 

70.  Rules. — 1.  To  find  the  least  common  multiple  of 
two  or  more  numbers  by  factoring,  Resolve  each  of  the  num¬ 
bers  into  its  prime  factors,  and  then  select  all  the  different  factors, 
talcing  each  the  greatest  number  of  times  it  is  found  in  any 
number.  The  product  of  the  different  factors,  thus  selected,  wiU 
be  the  least  common  multiple. 

2.  To  find  the  least  common  multiple  of  two  or  more 
numbers  by  division,  Write  the  numbers  in  a  line,  and  divide 
by  any  prime  divisor  of  two  or  more  of  them,  writing  the  quotients 
and  the  undivided  numbers  underneath.  Divide  these  resulting 
numbers  by  any  prime  divisor  of  two  or  more  of  them,  and  so 
proceed  until  no  two  of  the  resulting  numbers  have  a  common 
prime  divisor.  The  product  of  the  divisors  and  the  last  result¬ 
ing  numbers  will  be  the  least  common  multiple  required. 

Note. — If  no  two  of  the  given  numbers  have  a  common  divisor, 
their  product  will  be  the  least  common  multiple. 


FRACTIONS. 


43 


SECTION  VIII, 

FRACTIONS. 


HALVES  SIXTHS 

NUMERATION  AND  NOTATION. 


1.  If  an  apple  be  divided  into  two  equal  pieces,  what 
part  of  the  whole  will  one  piece  be? 

2.  If  an  apple  be  divided  into  four  equal  pieces,  what 
part  of  the  whole  will  one  piece  be?  Two  pieces?  Three 
pieces  ? 

3.  How  many  halves  in  a  single  thing  or  unit?  How 
many  fourths  ? 

4.  Which  is  the  greater,  one  half  or  one  fourth  of  a 
unit?  How  many  fourths  in  one  half? 

5.  What  is  meant  by  one  third  of  a  unit?  Two  thirds? 
One  sixth?  Three  sixths?  Two  fifths?  Four  fifths? 

71.  Such  parts  of  a  unit  as  one  half,  two  thirds,  three 
fourths,  etc.,  are  called  Fractions.  A  fraction  may  be  ex¬ 
pressed  in  figures  by  "writing  the  figure  denoting  the  number 
of  equal  parts,  into  which  the  unit  is  divided,  below  a  short 


44 


COMPLETE  ARITHMETIC. 


horizontal  line  (-g-),  and  the  figure  denoting  the  number  of 
equal  parts  taken,  above  the  same  line  (-&).  Thus,  f  ex¬ 
presses  five  sixths  of  a  unit. 

6.  What  does  |  express?  What  does  the  figure  7,  below 
the  line,  denote?  The  figure  5,  above  the  line? 

Read  the  following  fractions,  and  tell,  in  each  case,  what 
each  figure  denotes : 


7  & 

1  *  5 

10.  § 

13.  A 

Ifi  -9- 
2  0 

&  5 

O.  -g- 

11.  f 

14.  A 

17.  1? 

9.  $ 

12.  i 

15.  A 

18.  ff 

Write  the  following  fractions 

in  figures : 

(19) 

(20) 

(21) 

Two  fifths; 

Seven  twelfths; 

Twenty-four  fortieths ; 

Seven  ninths ; 

Ten  thirteenths , 

Thirty-five  fiftieths ; 

Ten  ninths. 

Twenty  seventeenths. 

Forty  fifty-fifths. 

22.  Is  the  fraction  f  greater 

or  less  than  1  ?  Why  ? 

23.  Is  |  greater  or  less  than 

a  unit  ? 

Why? 

Compare  the  value  of  each  of  the  following  fractions  with 

a  unit  or  1: 

24.  f 

26.  y 

28.  f 

30.  U 

25.  | 

27.  A 

29.  if 

31.  « 

32.  Deduce  from  the  above  examples  a  general  statement 
of  the  value  of  fractions  as  compared  with  a  unit  or  1. 


DEFINITIONS  AND  PRINCIPLES. 

72.  A  Fraction  is  one  or  more  of  the  equal  parts  of 
a  unit. 

The  unit  divided  is  called  the  Unit  of  the  Fraction  ;  and  one  of  the 
equal  parts,  into  which  it  is  divided,  is  called  a  Fractional  Unit.  An 
integer  is  composed  of  integral  units,  and  a  fraction  of  fractional 
units. 

73.  A  Common  Fraction  is  expressed  in  figures  by 
two  numbers,  one  written  over  the  other,  with  a  line  between 
them. 

Note.  —  Decimal  fractions  are  a  variety  of  common  fractions. 
(Art.  112.) 


$4 


FRACTIONS. 


cn 


n. 


45 


r 

r-  ‘ 

. 

The  number  above  the  line  is  called  the  Numerator;  and 
the  one  below  the  line,  the  Denominator. 


The  Denominator  of  a  fraction  denotes  the  number 
of  equal  parts  into  which  the  unit  is  divided. 

The  Numerator  of  a  fraction  denotes  the  number  of 
equal  parts  taken. 

The  numerator  and  denominator  are  called  the  Terms 
of  the  fraction. 


74.  Principle. — The  value  of  a  fraction  is  less  than  1 
when  its  numerator  is  less  than  its  denominator ;  equal  to  1  when 
its  numerator  equals  its  denominator ;  and  more  than  1  when  its 
numerator  is  greater  than  its  denominator. 

75.  Common  Fractions  are  Proper  or  Improper. 

A  Proper  Fraction  is  one  whose  numerator  is  less 
than  its  denominator ;  as,  f ,  f . 

An  Improper  Fraction  is  one  whose  numerator  is 
equal  to  or  greater  than  its  denominator. 

The  value  of  a  proper  fraction  is  less  than  one ;  and  the  value  of 
an  improper  fraction  is  equal  to  or  greater  than  one,  and  hence  it  is 
regarded  as  not  ■properly  the  fraction  of  a  unit. 


76.  Fractions  are  Simple,  Compound,  or  Complex. 

A  Simple  Fraction  is  a  fraction  not  united  with  an¬ 
other,  and  both  of  whose  terms  are  integral ;  as,  §. 

A  Compound  Fraction  is  a  fraction  of  a  fraction ; 
as>  f  §  5  i 

A  Complex  Fraction  is  one  having  a  fraction  in 

2  5  5  51 

one  or  both  of  its  terms ;  as,  g>  ~ 

4  5  -g-  H 

A  Mixed  Number  is  an  integer  and  a  fraction  united; 
as,  5£,  16£. 

77.  The  fraction  J-  may  be  considered  as  expressing  3 
fifths  of  1  unit,  or  1  fifth  of  3  units ;  and  hence  the  numer- 


46 


COMPLETE  ARITHMETIC. 


ator  of  a  fraction  may  denote  the  number  of  units  to  be 
divided,  and  the  denominator  the  number  of  parts  into 
which  the  numerator  is  to  be  divided.  Thus,  -f  may  be  read 
5  sixths,  or  1  sixth  of  5,  or  5  divided  by  6.  Hence, 

A  fraction  may  be  considered  an  expressed  division,  the 
numerator  being  the  dividend,  the  denominator  the  divisor,  and 
the  fraction  itself  the  quotient. 


REDUCTION  OF  FRACTIONS. 


Case  I. 

“Whole  or  Mixed.  TsTumlDers  reduced  to  Improper 

^Fractions. 

1.  How  many  thirds  in  an  apple?  In  4  apples?  7  ap- 
nles?  10  apples?  20  apples? 

2.  How  many  fifths  in  3  melons  ?  In  five  melons  ?  8 
melons?  12  melons?  15  melons? 

3.  How  many  sixths  in  1  ?  5?  8?  12?  20? 

4.  How  many  fourths  of  an  inch  in  2  and  1  fourth  inches  ? 
In  3|  inches?  6f  inches?  30^  inches? 

5.  How  many  fifths  in  3it?  4-f?  12-|?  16§? 

6.  How  many  tenths  in  5tV?  8t37?  12t\?  15^%? 


WRITTEN  PROBLEMS. 


7.  Reduce  225  to  sevenths. 


Process. 

225 
_ 7 

1575,  Ans. 
7 


225 1  to  sevenths. 

Process. 

225^ 

_ 7 

1580,  Ans. 

7 


8.  Reduce  324  to  ninths.  324J  to  ninths. 

9.  Reduce  4&J-J-  to  15ths.  65||  to  15ths. 

10.  Reduce  54^  to  20ths.  135|^  to  30ths. 

11.  Reduce  63  to  an  improper  fraction. 

12.  Reduce  74  J  A  to  an  improper  fraction. 

13.  Reduce  2063^  1°  an  improper  fraction. 


REDUCTION  OF  FRACTIONS. 


47 


14.  Reduce  145*  to  an  improper  fraction. 

Reduce  to  an  improper  fraction, 

15.  137*  17.  600£f  19.  208* 

16.  408*  18.  365*  20.  607* 

78.  Rules. — 1.  To  reduce  an  integer  to  a  fraction,  Mul¬ 
tiply  the  integer  by  the  given  denominator,  and  write  the  denomi¬ 
nator  under  the  product 

2.  To  reduce  a  mixed  number  to  a  fraction,  Multiply  the 
integer  by  the  denominator  of  the  fraction,  to  the  product  add 
the  numerator,  and  write  the  denominator  under  the  remit. 

Case  II. 

Improper  Fractions  reduced  to  Whole  or  Mixed 

Numbers. 

21.  How  many  dollars  in  8  half-dollars?  16  half-dollars? 
30  half-dollars? 

22.  How  many  pints  in  9  thirds  of  a  pint?  15  thirds  of 
a  pint  ?  33  thirds  of  a  pint  ? 

23.  How  many  days  in  20  fifths  of  a  day?  35  fifths  of  a 
day?  42  fifths  of  a  day? 

24.  Howt  many  units  in  36  ninths?  63  ninths?  75  ninths? 

25.  How  many  units  in  -2^?  *-?  -6^?  *-? 

26.  How  many  units  in  f -J  ?  fj?  -1*  ?  -2*3-  ? 

WRITTEN  PROBLEMS. 

27.  Reduce  -**•  to  a  whole  number. 

Process  :  *«=  256  -4-  16  =  16,  Ans. 

28.  Reduce  -2*3-  to  a  mixed  number. 

Reduce  to  a  whole  or  a  mixed  number, 


29. 

3  24 

12' 

<N 

CO 

6  3  6 

3T' 

35. 

1340 

38. 

w 

30. 

265 

it 

CO 

CO 

7  2  0 

TIT 

36. 

2  3  g  4 

39. 

31. 

TQJl 

1  6 

34. 

CO 

4  26  ft 
¥9 

40. 

4235  1 

~2T 

79.  Rule. — To  reduce  an  improper  fraction  to  an  integer 
or  mixed  number,  Divide  the  numerator  of  the  fraction  by  the 
denominator. 


48 


COMPLETE  ARITHMETIC. 


Case  III. 

Simple  Fractions  reduced  to  Lowest  Terms. 

41.  How  many  fourths  of  an  inch  in  2  eighths  of  an 
inch  ?  In  4  eighths  ?  6  eighths  ? 

42.  How  many  sixths  in  2  twelfths?  In  4  twelfths?  8 
twelfths?  10  twelfths? 

43.  How  many  sevenths  in  4  fourteenths?  In  6  four¬ 
teenths  ?  8  fourteenths  ?  12  fourteenths  ? 

44.  How  many  eighths  in  ^- ?  j§?  f§?  ff? 

45.  How  many  tenths  in  g^-?  f  §  ?  ff?  -ff? 

46.  Reduce  ff,  ff,  ff  each  to  fifths. 

47.  Reduce  -§f,  f-f ,  ff,  and  ff  each  to  sixths. 

48.  Divide  both  terms  of  -ff  by  3,  and  show  that  the  value 
of  the  fraction  is  not  changed. 

49.  Show  that  the  division  of  both  terms  of  any  fraction 
by  the  same  number  does  not  change  its  value. 


WRITTEN  PROBLEMS. 


50.  Reduce  fff  to  its  lowest  terms. 


105 


5 


140  i  5 


Process. 

21  21-4-7 

28  28-4-7 


~  105  -4-  35  3  . 

0r:  l40T35=i’  Am- 


Divide  both  terms  of  f|f  by  5,  re- 

3  ducing  it  to  \\ ;  then  divide  both 

4  terms  of  ff  by  7,  reducing  it  to  f. 
Since  |  can  not  be  reduced  to  smaller 
or  lower  terms,  it  is  in  its  lowest  terms. 
Or,  divide  both  terms  of  by  35, 


their  greatest  common  divisor,  reducing  the  fraction  to  f. 


Reduce  to  lowest  terms, 

51  8  1 

RA  264 

T so 

57.  m 

60. 

4  80 
1^45 

59  195 

55.  HI 

58.  A2* 

61. 

89  1 
T4F5 

216 

06.  yyy 

56.  m 

KQ  2  88 

TTTF 

62. 

63.  Express  the  quotient  of  195  divided  by  105  in  its 
simplest  form.  Am.  -If-. 

64.  Express  the  quotient  of  462  divided  by  441  in  its 

simplest  form.  128  —  256.  360  -4-  288. 


REDUCTION  OF  FRACTIONS. 


49 


65.  Express  the  quotient  of  576  -i-  432  in  its  simplest 
form.  216 -v-  324.  828 -i- 506. 


DEFINITIONS,  PRINCIPLE,  AND  RULES. 

80.  A  fraction  is  reduced  to  lower  terms  when  it  is  changed 
to  an  equivalent  fraction  with  smaller  terms. 

A  fraction  is  in  its  lowest  terms  when  its  terms  are  prime 
to  each  other. 

81.  Principle. — The  division  of  both  terms  of  a  fraction  by 
the  same  number  does  not  change  its  value. 

82.  Rules. — To  reduce  a  fraction  to  its  lowest  terms, 
1.  Divide  both  terms  of  the  fraction  by  any  common  divisor; 
then  divide  both  terms  of  the  resulting  fraction  by  any  common 
divisor ;  and  so  on,  until  the  terms  of  the  resulting  fraction  have 
no  common  divisor  except  1.  Or, 

2.  Divide  both  terms  of  the  fraction  by  their  greatest  common 
divisor. 


Case  IV. 


Fractions  reduced  to  Higher  Terms,  and  to  a  Com¬ 
mon  Denominator. 


66.  How  many  eighths  of  a  foot  in  1  fourth  of  a  foot?  In 
2  fourths?  3  fourths? 

67.  How  many  twelfths  in  3  sixths  ?  4  sixths  ?  5  sixths  ? 

68.  How  many  fifteenths  in  f  ?  -J  ?  £  ? 

69.  Change  f,  J,  and  §  each  to  twelfths. 

70.  Change  J,  f,  and  to  fortieths. 

71.  Change  f,  73^,  and  to  sixtieths. 

72.  Change  f,  f,  and  to  thirtieths. 

73.  Change  f,  f,  and  to  fortieths. 

74.  Multiply  both  terms  of  §  by  4,  and  show  that  the 
value  of  the  fraction  is  not  changed. 

75.  Show  that  the  multiplication  of  both  terms  of  any 

fraction  by  the  same  number  does  not  change  its  value. 
C.Ar. — 5. 


50 


COMPLETE  ARITHMETIC. 


WRITTEN  PROBLEMS. 

76.  Reduce  -J,  f,  and  to  equivalent  fractions  having  a 
common  denominator. 


Process. 


5 

■6 

20 

zr 


2 

8 

21 

2? 


1  1 

TZ 

2  2 
Zf 


Reduce  the  fractions  to  twenty-fourths,  thus :  f  = 


20  . 
Z¥’ 


1 


2  1  . 
2?  J 


11- 
TZ  — 


2  2 
2 1* 


Reduce  to  a  common  denominator, 


77. 

78. 

79. 


2 

3’ 

3 

¥’ 

2 

T* 


3 
5  ’ 
5 
8 ’ 

5 

T¥> 


5 

6 
7 

1  2 
_1_ 
2 


80. 

81. 

82. 


3 

4  ’ 
1 

8’ 

1 

Tf’ 


o_ 

8’ 

5. 

6> 

3 

4> 


7 

1  6 
7 
9 

i 


83. 

84. 

85. 


3 

S’ 

4 
5’ 
2 

3’ 


5 

6  ’ 
9_ 

1  0’ 
3 

T> 


J_3 
2  0’ 
5 
6’ 


1  1 

TU 

1  1 
¥TT 

4 

2  1 


86.  Reduce  f ,  T7^-,  and  f  J  to  equivalent  fractions  hav¬ 
ing  the  least  common  denominator. 


Process. 

5 

7 

1  1 

2  1 

J 

TZ 

2? 

3Z 

60 

42 

44 

63 

Z6 

¥6 

ZZ 

96 

The  least  common  multiple  of  8,  16,  24,  and 
32  is  96,  and  hence  96  is  the  least  common  de¬ 
nominator.  Change  the  fractions  to  96ths. 


5  6  0.  7_ 

J Z6  >  T6 


4  2  . 
96  > 


1  1 

Z¥ 


44  .  21  -  6  3 

Z  6  >  3  2  THT  * 


Reduce  to  the  least  common  denominator, 


00 

• 

4 

9’ 

1 1 

1  2’ 

1  7 
3S’ 

If 

90. 

7 

1  2’ 

8 

2  1’ 

1  1 

28’ 

1  7 

4  2 

Oo 

oo 

h 

3 

S’ 

5 

S’ 

i 

91. 

2 

5’ 

1  1 
12’ 

1  1 

10’ 

29 

3¥» 

3  1 

6~0 

89. 

10’ 

8 

1  5’ 

1  1 

2  0’ 

1  3 

3¥ 

92. 

5 

9’ 

1  4 
3S’ 

2  2 
45’ 

1  2 

6  3’ 

1  0  1 
TIT 

DEFINITIONS,  PRINCIPLE,  AND  RULES. 

83.  A  fraction  is  reduced  to  higher  terms  when  it  is 
changed  to  an  equivalent  fraction  with  greater  terms. 

84.  Several  fractions  are  reduced  to  a  Common  Denomi¬ 
nator  when  they  are  changed  to  equivalent  fractions  with 
the  same  denominator. 

When  the  common  denominator  of  several  fractions  is 
the  smallest  denominator  which  they  can  have  in  common, 
it  is  called  their  Least  Common  Denominator . 

85.  Principle. — The  multiplication  of  both  terms  of  a  frac¬ 
tion  by  the  same  number  does  not  change  its  value. 


REDUCTION  OF  FRACTIONS. 


51 


86,  Rules. — 1.  To  reduce  a  fraction  to  higher  terms, 
Divide  the  given  denominator  by  the  denominator  of  the  fraction, 
and  multiply  both  terms  by  the  quotient. 

2.  To  reduce  fractions  to  the  least  common  denominator, 
Divide  the  least  common  multiple  of  the  denominators  by  the  de¬ 
nominator  of  each  f  raction,  and  multiply  both  terms  by  the  quotient . 


Case  V. 

Compound  Fractions  reduced  to  Simple  Fractions. 

93.  How  much  is  1  half  of  1  third  of  a  pear  ?  1  half  of 
1  fourth  of  a  pear? 

94.  A  father  divided  \  of  a  pine-apple  equally  between  3 
boys :  what  part  of  the  pine-apple  did  each  boy  receive  ? 

95.  What  is  £  of  ^  ^  of  4  ?  ^  of  £  ? 

96.  What  is  \  of  |-?  \  of  f  ? 

97.  What  is  -J-  of  ^  ?  4  of  f  ?  §  of  f  ? 

98.  What  is  y  of  i ?  7  of  |?  f  of  |  ? 

99.  What  is  |  of  f?  §  of  -§-?  •§  of  -J? 

100.  What  is  |  of  -f?  f  of  f  ?  f  of  t7j? 

101.  What  is  i  of  12?  i  of  12£?  }  of  13*  ? 

Solution.— 4  of  of  12,  which  is  4,  -f  }  of  or  §,  which  is 

$  or  4  -|-  \  —  4|.  Hence,  |  of  13|  is  4 

102.  What  is  \  of  17-i?  i  of  21|?  \  of  33^? 

103.  What  is  |  of  12?  |  of  12£?  f  of  I64? 

104.  What  is  f  of  22^  ?  f  of  25J  ?  i  of  37£? 

105.  What  is  |  of  33£?  £  of  42£?  ^  of  62£? 

WRITTEN  PROBLEMS. 


106.  Reduce  §  of  of  7-J-  to  a  simple  fraction. 

Process. 

5  0f  _3_  of  74  =  ^  —  2^.  —  5,  Ans. 

9  10  2  9  X  15  X  2  270  6 

Or:  4  of  Tsj  of  7|  =  £X_J_X_£$ _ 5 

'  9X15X  2  6 

3 


52 


COMPLETE  ARITHMETIC. 


Reduce  to  a  simple  fraction, 


107.  |  of  J  of  f 

108.  f  of  }  of  2J 

109.  f  of  ji  of  ll- 
HO.  f  of  f  of  2| 


111. 

112. 

113. 

114. 


of  f  of  f  of  3i 


j  of 


of  j-  of  4f 


3 

5  9  7  ,J  3 

_4 

1  of  f  Of  2f  Of  2| 
f  of  y9u  of  yy  of  3|- 


87.  Rules. — To  reduce  a  compound  fraction  to  a  simple 
fraction,  1.  Multiply  the  numerators  together  for  a  numerator , 
cmd  the  denominators  together  for  a  denominator.  Or, 

2.  Indicate  the  continued  midtiplication  of  the  numerators , 
and  also  of  the  denominators,  and  reduce  the  resulting  fraction 
to  its  lowest  terms  by  cancellation. 


REVIEW  PROBLEMS. 

115.  Reduce  16  to  a  fraction  having  8  for  a  denominator. 

116.  Change  35 -=-21  to  a  fraction  in  its  lowest  terms. 

117.  How  many  15ths  of  a  gallon  in  33|  gallons? 

118.  Reduce  \°-£  to  a  mixed  number  with  the  fraction  in 
its  lowest  terms. 

119.  Reduce  ,  and  $7T°T°-  each  to  whole  or 

mixed  numbers. 

120.  Reduce  §,  T7y,  and  ||  each  to  30ths. 

121.  Reduce  12|,  18J,  and  33 J  each  to  12ths. 

122.  Reduce  T2-  of  of  2^3  of  24  to  a  simple  fraction. 

123.  Reduce  |,  and  ^  to  a  common  denominator ;  to 
the  least  common  denominator. 

124.  Reduce  f,  5|-,  and  f  of  f  to  their  least  common 
denominator. 

Suggestion. — First  reduce  5|  and  of  f  to  simple  fractions. 

125.  Reduce  f,  f  of  f,  and  -f  of  6|  to  their  least  common 
denominator. 

126.  Reduce  |  of  2-|,  of  3,  and  ^  of  -J  of  6y  to  their 
least  common  denominator. 

127.  Reduce  -J  of  |,  §  of  2-J,  and  -J  of  13|  to  their  least 
common  denominator. 

128.  Reduce  -f-  of  5|,  2J  of  34,  and  2T3y  to  their  least 
common  denominator. 


ADDITION  OF  FRACTIONS. 


53 


ADDITION  OF  FRACTIONS. 

1.  A  clerk  spends  f  of  his  salary  for  board,  -§  of  it  for 
clothing,  and  %  for  other  expenses :  what  part  of  his  salary 
does  he  spend? 

2.  How  many  ninths  in  J,  •§,  and  ^? 

3.  A  man  traveled  ^  of  his  journey  the  first  day,  and  \ 
of  it  the  second  day :  what  part  of  the  journey  did  he  travel 
in  the  two  days? 

4.  How  many  twelfths  in  J  and  ^?  \  and  ^-? 

5.  A  owns  f  of  a  vessel,  and  B  f  of  it :  what  part  of  the 
vessel  do  both  own  ? 

6.  What  is  the  sum  of  J-  and  |?  f  and  f? 

7.  §  and  ^?  f  and  y^?  -§  and  -J?  and  J? 

8.  -f  and  ^  and  J  ?  j-  and  T97  ?  f  and  |  ? 

9.  ^  and  5J?  2J  and  6J?  5^-  and  6J?  8J  and  9f  ? 

Suggestion. — First  add  the  fractions  and  then  the  integers. 

10.  Show  that  fractions,  having  a  common  denominator, 
express  like  fractional  units,  and  that  only  like  fractional 
units  can  be  added. 


WRITTEN  PROBLEMS. 


11.  What  is  the  sum  of  yf>  and  ? 

Process :  -1  4-  1A  4- 1®  —  9+16  +  10  —  35  —  112  ^ns. 

23  1  23  1  23  23  -  23  23 

12.  What  is  the  sum  of  -ff,  |-J,  and  T^? 

13.  What  is  the  sum  of  f§,  J-^,  and  f  J? 

14.  What  is  the  sum  of  Ty^,  and  ? 

15.  What  is  the  sum  of  f,  T72,  and  -1-J-? 


Process. 


f  +  T2  +  TS  — 

H  +  tt  +  *i  = 
H=  Hi  Am. 


Since  unlike  fractional  units  can  not  be  added, 
reduce  the  fraetions-f*  -fa,  and  H  1°  a  common 
denominator,  and  then  add  the  resulting  frac¬ 
tions. 


54 


COMPLETE  ARITHMETIC. 


16. 

17. 

18. 

19. 

20. 


Add  f  and 

and  yq. 

ib  and 
and 


3. 

8’ 


1  3 
1  8  • 

1  3 
1  5* 
31 
4  2 ' 


_o 

6  ’ 

3 

5 ’  1  0 ’ 

A*  ib  and 

26.  Add  J,  f  of  f,  and  f  of  f  of  2J. 


21. 

22. 

23. 

24. 

25. 


9 

1  0’ 


9  2  3 

2ir>  j  o  > 


4  1 
Q~U- 


o 
6  ’ 


A 

9’ 


and 

and  tJ* 

1  3  ±5  J2.0  o-rifl  2 

IF’  2  7’  8  f >  ,uu  3* 

1  _7_  _7  anfl 

8’  12’  18’  ^4  * 

and 


i 

12’ 


_1_ 

1  5’ 


1 

28"’ 


1_ 

■JO' 


Process. 


2nf  3  —  2 
?  OI  x  —  x 


of 


of  }  = 


Since  § 


of  f  =  f,  and  f  of  | 
of  2£-  =  f ,  the  sum  of  f  -f-  §  of 


31215  -  30116125  -  131 

1  +  5  +  J  —  loTJo  Tfo  —  -*-40 


f  +  y  of 


of  2}=}  +  }  +  }. 


27.  Add  |  of  |,  |  of  fg-  of  2J,  and  J. 

28.  Add  J  of  2^-,  f  of  J,  and  f  of  J  of  6. 

29.  Add  t34,  y  of  5,  and  -f-  of  f  of  -§. 

30.  Add  A-  of  2,  §  of  f,  4  of  4  of  T5o,  and  3k 

31.  Add  33},  37},  55 J,  and  66f. 


Process. 

33|  x\ 

37  \  xx 

i=;r:  3  9 

oo¥  x5- 

°°3  TS 

193x,  Ans. 


The  sum  of  33^,  37|,  55|,  and  66§,  equals  the  sum 
of  i  +  y  +f  +  f  added  to  the  sum  of  33  -f  37 -(-  55  -f-  66. 
4  +  x  +  f  +  f  =  2x3j  or  2|.  Write  the  \  under  the  frac¬ 
tions  and  add  the  2  with  the  integers.  The  sum  is 
193i. 


32.  Add  39},  56J,  88},  and  104ft. 

33.  Add  45,  87f,  66§,  and  7 5}. 

34.  Add  121  i6f,  18J,  30},  33J,  and  62}. 

35.  Add  -§,  f  of  f,  16f,  and  48 J. 

36.  Add  $5.12},  $3.18},  $8.25,  and  $3.81}. 

37.  Add  },  ft,  }  of  5},  and  65ft. 

38.  Add  },  },  J-,  ft,  and  ft  of  2|. 


PRINCIPLES  AND  RULES. 

88.  Principles. — 1.  Only  like  fractional  units  can  be  added. 
Hence, 

2.  Fractions  must  have  a  common  denominator  before  they  can 
be  added. 


SUBTRACTION  OF  FRACTIONS. 


55 


89.  Rules. — 1.  To  add  fractions,  Reduce  the  fractions  to  a 
common  denominator,  add  the  numerators  of  the  new  fractions , 
and  under  the  sum  write  the  common  dejiominator. 

2.  To  add  mixed  numbers,  Add  the  fractions  and  the  in¬ 
tegers  separately ,  and  combine  the  results. 

Notes. — 1.  Compound  fractions  must  be  reduced  to  simple  fractions 
before  they  can  be  added. 

2.  When  mixed  numbers  are  small  they  may  be  reduced  to  inn 
proper  fractions  and  then  added. 


SUBTRACTION  OF  FRACTIONS. 

1.  A  boy  spent  f-  of  his  money  for  a  slate:  what  part  of 
his  money  has  he  left? 

2.  How  much  is  f  less  §  ?  f  less  f  ?  f  less  f  ? 

3.  How  much  is  less  less  T52?  less  T92? 

4.  A  bought  f  of  a  bushel  of  clover  seed  and  sold  ^  of  a 
bushel  to  B :  what  part  of  a  bushel  has  A  left  ? 

Suggestion. — Change  f  and  ^  to  twelfths. 

5.  How  much  is  J  less  |  ?  f  less  §  ?  f  less  \  ? 

6.  f  less  f  ?  f  less  -§  ?  ^  less  J  ?  f  less  |  ? 

7.  ^  less  f  ?  §  less  f  f  less  f  ?  §  less  y  ? 

8.  |  less  \  ?  J  less  f  ?  ^  less  f  ?  j  less  ^  ? 

9.  ^  less  }  ?  yj  less  J  ?  f  less  ?  f  less  f  ? 

10.  5£  less  3|  ?  6|  less  4^?  9f  less  7£?  12^  less  6^? 

1 1 .  Why  can  not  f  be  subtracted  from  f  without  first  re¬ 
ducing  the  fractions  to  a  common  denominator  ? 


WRITTEN  PROBLEMS. 


12.  Subtract 


1  9  fpnm  2  7 
TS  Irom  To- 


Process  : 


2  7 
35 


li 

35 


27  —  19 
35 


=  A  Jns- 


13.  Subtract  from  |«;  from  Jf;  ||  from 

14.  Subtract  yC-  from  -JJ. 

Process  :  =  f  §  —  f §  =  ,  Ans. 


56 


COMPLETE  ARITHMETIC. 


How  much  is 


15. 

1 4  _ 

- v 

18. 

33  _ 

_  1 1  ? 

21. 

1 0  _ 

¥7 

-  9  ? 
47  - 

16. 

1  3 

T7 

-A? 

19. 

1 9  _ 
¥7 

-11? 

T  ¥  • 

22. 

23  _ 
¥( J 

_  11  ? 
54  ’ 

17. 

-A? 

20. 

-i — 
1 2 

4  ? 

“  T7  • 

23. 

29  _ 
70 

1  3  ? 
45  ’ 

24.  From  J  of  f  take  \  of  J  of  f. 


Process  :  |  of  f  —  f  f  of  f  of  f  =  |  f  —  i  =  Ans. 


25.  From  f  of  f  take  §  of  -§  of  -§. 

26.  From  f  of  7  take  %  of  f  of  7. 

27.  From  §  of  £  of  f  take  f  of  J  of  f. 

28.  From  -f  of  f  of  2 ^  take 

29.  From  340 J  take  247|. 


Process. 
340f  * 

247|  M 
92^,  Ans. 


First  subtract  the  fractions  and  then  the  integers. 
Since  is  greater  than  -fa,  add  §§  to  making  §§, 
and  then  subtract  from  §§,  writing  the  difference, 
If,  under  the  fractions,  and  adding  1  (|f)  to  the  7 
units  before  subtracting  the  integers. 


30.  93J  —  46J  =  ?  33.  241  J  —  153yV  =  ? 

31.  56|  —  37|-  =  ?  34.  $2.33J— $1.62i  =  ? 

32.  108f— 90f  =  ?  35.  $3.12|  — $2.48§=? 

36.  What  fraction  added  to  f  will  make  -j-^? 

37.  What  number  added  to  6f  will  make  16|? 

38.  From  the  sum  of  §  and  |  take  their  difference. 

39.  From  f  + -f  take  J-  —  -J.  ~~  ' 

40.  From  f  ~h  |  +  t7o  ta^e  f  °f  1£* 

41.  From  J  -J-  f  take  ^  —  §  of  §. 

42.  From  a  cask  containing  45^  gallons  of  sirup,  a  grocer 
sold  one  customer  16f  gallons  and  another  21f  gallons:  how 
many  gallons  remained  unsold  ? 

43.  A  man  bequeathed  of  his  property  to  his  wife,  T5^ 
of  it  to  his  children,  and  the  remainder  to  a  college  for  its 
better  endowment.  What  part  of  his  property  did  the  col¬ 
lege  receive? 

44.  A  man  owning  •§  of  a  factory,  sold  §  of  his  share : 
what  part  of  the  factory  did  he  still  own? 


MULTIPLICATION  OF  FRACTIONS. 


57 


45.  Two  ninths  of  a  pole  is  in  the  mud,  f-  of  it  in  the 
water,  and  the  rest  of  it  in  the  air:  what  part  of  the  pole  is 
in  the  air  ? 

46.  The  part  of  a  pole  broken  off  by  the  wind  was  f  of 
the  whole  pole,  and  f  of  the  part  still  standing  was  above 
the  ground:  what  part  of  the  pole  was  in  the  ground? 

PRINCIPLES  AND  RULES. 

90.  Principles. — 1.  The  minuend  and  subtrahend  must  de¬ 
note  like  fractional  units.  Hence, 

2.  Fractions  must  have  a  common  denominator  before  their 
difference  can  be  found. 

u  ■  — 

91.  Rules. — 1.  To  subtract  fractions,  Reduce  the  fractions 
to  a  common  denominator ,  subtract  the  numerator  of  the  subtra¬ 
hend  from  the  numerator  of  the  minuend ,  and  under  the  differ¬ 
ence  write  the  common  denominator. 

2.  To  subtract  mixed  numbers,  Subtract  first  the  fractions , 
and  then  the  integers ,  and  unite  the  results. 

Notes. — 1.  Compound  fractions  must  be  reduced  to  simple  fractions 
before  they  can  be  subtracted. 

2.  When  mixed  numbers  are  small  they  may  be  reduced  to  im¬ 
proper  fractions,  and  then  subtracted. 

MULTIPLICATION  OF  FRACTIONS. 

Case  I. 

Fractions  multiplied,  "by  Integers. 

1.  How  much  is  twice  2  ninths  of  an  inch?  4  times  2 
ninths  of  an  inch? 

2.  If  a  basket  hold  f  of  a  bushel,  how  many  bushels  will 
8  baskets  hold?  10  baskets? 

3.  How  much  is  8  times  f  ?  10  times  f  ?  20  times  f  ? 

4.  6  times  f?  8  times  J?  9  times  ^?  12  times  -J? 

5.  7  times  {-£?  9  times  T4-?  8  times  ^<5?  U  times  §? 

6.  6  times  5^?  9  times  6J?  7  times  12-J-?  10  times  7-f? 

7.  8  times  12^-?  6  times  16J?  5  times  33^?  7  times 

30i  ? 


58 


COMPLETE  ARITHMETIC. 


8.  Why  does  multiplying  the  numerator  of  T\  by  3 
multiply  the  fraction  by  3  ? 

9.  Why  does  dividing  the  denominator  of  ^  by  3  mul¬ 
tiply  the  fraction  by  3  ? 

10.  In  how  many  ways  may  a  fraction  be  multiplied  by 
an  integer? 


WRITTEN  PROBLEMS. 


Multiply 

11.  by  9. 

12.  if  by  10. 

13.  by  24. 

14.  H  by  45. 


15.  -fa  by  12. 

16.  by  16. 

17.  by  60. 

18.  Hi  by  25. 


19.  62J  by  36. 

20.  45|  by  80. 

21.  $5.18f  by  32. 

22.  $66f  by  52. 


PRINCIPLE  AND  RULES. 

92.  Principle. — A  fraction  is  multiplied  by  multiplying  its 
numerator  or  dividing  its  denominator. 

93.  Rules. — 1.  To  multiply  a  fraction  by  an  integer, 
Multiply  the  numerator  or  divide  the  denominator. 

2.  To  multiply  a  mixed  number  by  an  integer,  Multiply 
the  fraction  and  the  integer  separately,  and  add  the  products. 


Case  II. 

Integers  multiplied  by-  Fractions. 

23.  If  a  ton  of  hay  cost  $16,  what  will  f  of  a  ton  cost? 
f  of  a  ton  ? 

24.  If  an  acre  of  land  is  worth  $50,  what  is  i  an  acre 

worth  ?  |  of  an  acre  ? 

25.  What  is  i  of  42?  f-  of  42?  £  of  42? 

26.  What  is  f  of  56  ?  f  of  56  ?  J  of  56  ? 

27.  -f  of  63?  *  of  84?  H  of  99?  $  of  56? 

Solution. —  ^  of  56  =  6§,  and  |  of  56  =  7  times  6§  =  43§. 

28.  |  of  66?  i  of  66?  H  of  66?  f  of  74? 


MULTIPLICATION  OF  FRACTIONS. 


59 


29.  What  is  16  xf?  50  Xf?  42  Xf? 

Solution. — Since  f  =  §  of  1,  16  X  f  =  f  of  16  X  1  —  I  of  16  =  12. 

30.  57  Xf?  75X|?  87  X  *?  95  X*?  76  X|? 

31.  47  X|?  68  Xi?  75  X  f?  83  X*?  100  X  A? 

32.  Show  that  the  product  of  an  integer  by  a  fraction 
equals  the  fraction  of  the  integer. 

WRITTEN  PROBLEMS. 

33.  Multiply  654  by  . 

Process. 


12  )  654  654  Since  Tr2  =  7  times  or  of  7, 

54j  Qr .  7  .  the  product  of  654  Xfi=7  times 

7  12  )  4578  TJ2  0f  654^  or  ^  0f  7  times  654. 


3811,  Ans. 

381 

1 

2 

34.  66  by 

1 

9  • 

37. 

784 

by 

1 3 

40* 

40. 

757 

by 

1  T* 

35.  58  by 

1  3 

TT- 

38. 

648 

by 

2  5 

2  6* 

41. 

908 

by 

}  of  21 

36.  92  by 

7 

22* 

39. 

564 

by 

3  3 
40* 

42. 

588 

by 

T5  °f 

43.  Multiply  256 

by  27£. 

406  by 

33f. 

Suggestion. — Since 

256  X  27$ 

256  x 

27  + 

256  X  f , 

first  mul 

tiply  by  the  integer  and  then  by  the  fraction,  and  add  the  products. 


44.  66  by  8}.  47.  645  by  12 J.  50.  745  by  60|. 

45.  72  by  9f.  48.  465  by  18f.  51.  385  by  45x4 

46.  96  by8T%.  49.  406  by  33 J.  52.  708  by  60f. 

PRINCIPLE  AND  RULES. 

94.  Principle. — The  product  of  an  integer  by  a  fraction 
equals  the  fraction  of  the  integer.  Thus,  5X|  =  t  of  5. 

95.  Rules. — 1.  To  multiply  an  integer  by  a  fraction, 
(1)  Divide  the  integer  by  the  denominator,  and  multiply  the 
quotient  by  the  numerator.  Or,  (2)  Multiply  the  integer  by  the 
numerator ,  and  divide  the  product  by  the  denominator. 

2.  To  multiply  an  integer  by  a  mixed  number,  Multiply 
by  the  integer  and  the  fraction  separately,  and  add  the  products. 


60 


COMPLETE  ARITHMETIC. 


Case  III. 

Fractions  multiplied  ‘by  UTractions. 

53.  What  is  }  of  1?  }  of  J?  -|  of  f  ? 

54.  |  of  |?  |  off?  f  of  }?  f  of  f? 

55.  What  is  f  X  f  ?  f  X  f  ?  |Xf? 

Suggestion. —  f  X  f  =  f  of  f ;  |  X  y  =  I  of  f,  etc. 

56.  fX*t  fX}}?  }X*? 

57.  What  is  }  of  12}?  }  of  13}?  }  of  13}. 

Solution. —  |  of  13^  =  j  of  12  +  1  °f  1|  =  4  +  A  —  4jz  >  add  I 
of  131  —  ^  times  4r\  =  8f. 

58.  f  of  16f  ?  f  of  22}?  f  of  42}?  ^  of  62}? 

59.  }  of  37}?  f  of  42} ?  }  of  65}?  -fy  of  100}? 

60.  Show  that  f  X  f-  =  f  of  f. 


WRITTEN  PROBLEMS. 


61.  Multiply  -}§  by  f. 


II X 


Process. 

3  — 
T  — 

13  X  3 _ I  ., 

15X4 

Ans. 

62. 

13  Ev  11 

nr  Dy  T3* 

66. 

63. 

10  Ev  2  2 

TT  °y  2T- 

67. 

64. 

19  Ev  15 

F5  °y  3¥' 

68. 

65. 

10  Ev  3  9 

if  °y  nr- 

69. 

Since  f  =  }  of  3,  the  product  of 
rt  X  f  =  1  of  3  times  }f  =  }  of 
13  X  3  _  13  X  3 
15  15  X  4 


1  3 
SO* 


f  by  i  of  £• 
£  °f  A 
£by| 


by  A- 
by  £• 

2i  by  A  of 


70.  2}  by  3}. 

71.  4}  by  5}. 

72.  6}  by  3f. 

73.  10}  by  2-J. 


74.  What  will  f  of  a  yard  of  cloth  cost  at  $f  a  yard? 
At  a  yard ? 

75.  What  will  5}  pounds  of  flour  cost  at  4}  cents  a 
pound  ?  At  61  cents  a  pound  ? 

76.  What  will  2}  pounds  of  tea  cost  at  $lf  a  pound? 
At  $1}  a  pound  ? 

77.  What  is  the  cost  of  35  barrels  of  flour  at  $6}  a 
barrel?  At  $7}  a  barrel? 


DIVISION  OF  FRACTIONS. 


61 


78.  A  man  owned  ^  of  a  ship  which  was  sold  for  $13250 : 
what  was  his  share  of  the  money? 

79.  What  is  the  product  of  §  of  2 \,  f  of  -fa  of 
and  2-g-  ? 

80.  What  will  12-J-  pounds  of  butter  cost  at  18f  cents  a 
pound  ?  At  22^  cents  a  pound  ? 


PRINCIPLE  AND  RULES. 


96.  Principle.  —  The  product  of  a  fraction  by  a  fraction 
equals  the  fraction  of  the  fraction.  Thus,  f  X  f  =  f  of  f . 

97.  Rules. — 1.  To  multiply  a  fraction  by  a  fraction, 
Multiply  the  numerators  together,  and  also  the  denominators. 

2.  To  multiply  a  mixed  number  by  a  mixed  number, 
Reduce  the  mixed  numbers  to  improper  fractions,  and  proceed 
as  above. 


Notes.— 1.  Mixed  numbers  may  be  multiplied  together  by  first  mul¬ 
tiplying  the  integers  ;  next  multiplying  each  integer  by  the  fraction  united 
with  the  other  integer ;  next  multiplying  the  two  fractions  ;  and  then  add¬ 
ing  the  four  products.  Thus,  18§  X  12^  =  18  X  12  +  18  X  \  +  12  X 
f  +  |  X  hut  in  most  cases  it  is  shorter  to  reduce  the  mixed 
numbers  to  improper  fractions. 

2.  Cases  I  and  II  may  be  included  in  Case  III,  by  changing  the 
integer  to  the  form  of  a  fraction.  Thus,  f  X  5  =  f  X  f>  and  8  X  f  = 

8  v  3 

T  A  z- 

3.  The  process  of  multiplying  fractions  may  be  shortened  by  can¬ 
cellation.  Compound  fractions  need  not  be  reduced  to  simple  frac¬ 
tions,  since  f  X  I  of  If  =  f  X  1  X  rr  • 


DIVISION  OF  FRACTIONS. 

Case  I. 

Fractions  divided  Toy  Integers. 

1.  If  a  man  can  do  of  a  piece  of  work  in  3  days,  how 

much  can  he  do  in  1  dav? 

%/ 

2.  A  man  divided  f  of  a  farm  equally  between  4  sons : 
what  part  of  the  farm  did  each  receive? 

3.  If  5  yards  of  muslin  cost  £  of  a  dollar,  what  will  1 
yard  cost? 


62 


COMPLETE  ARITHMETIC. 


4.  If  10  oranges  cost  -§  of  a  dollar,  what  will  1  orange 
cost? 

5.  If  8  bushels  of  oats  cost  $2§,  what  will  1  bushel  cost  ? 

6.  If  ff  of  a  melon  be  divided  into  5  equal  parts,  what 
will  each  part  be? 

7.  Why  does  dividing  the  numerator  of  f  by  4  divide 
the  fraction  by  4? 

8.  Why  does  multiplying  the  denominator  of  f  by  4 
divide  the  fraction  by  4? 

9.  In  how  many  ways  may  a  fraction  be  divided  by  an 
integer  ? 

WRITTEN  PROBLEMS. 


10.  Divide  by  6. 


1  2 

2l>  •  ° 


Process. 

12 -f- 6 


25 


=  &,  Ans- 


Or:  if -+-6 


12 


25  X  6 


2 

2  V 


Giyipp  1 2  *  “I  ■  12  12 

oince  ■  -L  —  ss, 


1  2 


2S 
12  +  6 

25  ’ 


or 


12 


25X6 


+  6  =  f  of 
Or,  since  to 


divide  a  number  by  6  is  to  find  f  of 


of  it,  if  6  —  e 

12 


iof  if 


12  -f-  6 


25 


-,  or 


25X6 


Divide 


11.  ff  by  8. 

12.  -if  by  7. 

13.  ff  by  11. 


14.  if  by  15. 

15.  if  by  20. 

16.  iff  by  25. 


17.  2f  by  8. 

18.  5-J  by  12. 

19.  6J-  by  10. 


PRINCIPLE  AND  RULES. 

98.  Principle. — A  fraction  is  divided  by  dividing  its  nu¬ 
merator  or  multiplying  its  denominator. 

99.  Rules. — 1.  To  divide  a  fraction  by  an  integer,  Di¬ 
vide  the  numerator  or  multiply  the  denominator. 

2.  To  divide  a  mixed  number  by  an  integer,  (1)  Reduce 
the  mixed  number  to  an  improper  fraction  and  divide  as  above ; 
or,  (2)  Divide  the  integral  part  and  then  the  fraction ,  and  unite 
the  quotients. 


DIVISION  OF  FRACTIONS. 


63 


Case  II. 

Integers  divided  by  Fractions. 

20.  How  many  times  is  -J  of  a  cent  contained  in  4  cents? 

Solution. — In  four  cents  there  are  20  fifths  of  a  cent,  and  2  fifths 

of  a  cent  are  contained  in  20  fifths  of  a  cent  10  times. 

- - . - 

2x.  If  a  fruit  jar  hold  f  of  a  gallon,  how  many  jars  will 
hold  6  gallons?  12  gallons?  18  gallons? 

22.  If  f  of  a  yard  of  silk  will  make  a  vest,  how  many 
vests  will  5  yards  make?  7  yards?  10  yards? 

23.  If  a  yard  of  cloth  cost  $f,  how  many  yards  can  be 
bought  for  $10?  For  $15?  For  $20? 

24.  How  many  times  is  J  contained  in  8?  J  in  12?  §  in 
9?  |  in  15?  |  in  9?  £  in  12? 

25.  How  many  times  is  |  contained  in  12?  -J  in  15? 

26.  Show  that  8  ~~  f  =  . 

o 

WRITTEN  PROBLEMS. 

27.  What  is  the  quotient  of  25  -s-  £  ? 

Process  :  25  -5-  £  =  — ^  =  28f,  Ans. 

Note. — It  will  be  noticed  that  the  integer  is  multiplied  by  the 
denominator  of  the  fraction  and  the  product  divided  by  its  nu¬ 
merator. 

What  is  the  quotient  of 

28.  21 -j-*?  31.  100 -f-|^?  34.  75-r-6£? 

29.  42-s-tt?  32.  96--££?  35*  120  -*-3£? 

30.  72  -8- If?  33-  125-*- ft?  36.  225 -s- 5}? 

100.  Rules. — To  divide  an  integer  by  a  fraction,  1.  Mul¬ 
tiply  the  integer  by  the  denominator  of  the  fraction,  and  divide 
the  product  by  the  numerator .  Or, 

2.  Divide  the  integer  by  the  numerator,  and  multiply  the  quo¬ 
tient  by  the  denominator. 


64 


COMPLETE  ARITHMETIC. 


Case  III. 

Fractions  divided,  "by-  Fractions. 


37.  How  many  times  is  -f  of  an  inch  contained  in  |  of  an 
inch  ?  -J  of  an  inch  in  -f  of  an  inch  ? 

38.  How  many  times  f  in  f  ?  -f  in  -|  ?  J-  in  ^°-  ? 

39.  How  many  times  -f  in  -J?  |  in  |  in  4^?  tj-  in 

¥?  A  m*?  *inH» 

40.  How  many  times  is  |  contained  in  |  ?  |  in  -§? 
Suggestion. — Change  the  fractions  to  twelfths. 

41.  How  many  times  -§  in  -f?  f  in  f-  ?  -J  in  -J?  |  in  T\? 

49  in  4  9  3.  in  _7_  ?  __3_  in  14?  2  in  3  ?  2.  in  A? 

“iZ.  i  0  m  "o  •  8  1  2  •  1  0  111  T  5  •  3  111  7  *  5  111  8  * 

43.  Show  that  the  quotient  of  two  fractions  having  a  com¬ 

mon  denominator,  equals  the  quotient  of  their  numerators. 


WRITTEN  PROBLEMS. 


44.  Divide  J 


Process  :  ,1- 


Since 


7X5 


»  and  | 


3X8 


\ 


7 
>  8 


3 

7 


7X5  .  3  X  8  _  7  X  5 


40  a  40  ’ '  °  “  40  40  3X8' 

It  is  thus  seen  that  inverting  the  terms  of  the  divisor,  and  taking  the 
product  of  the  numerators  for  the  numerator,  arid  the  product  of  the 
denominators  for  the  denominator,  is  the  saihe  as  reducing  the  frac¬ 
tions  to  a  common  denominator,  and  dividing  the  numerator  of  the 
dividend  by  the  numerator  of  the  divisor. 


7X5 

Note. — That  |  -4-  f  =  n  may  also  be  thus  explained :  f 


times  |,  and  since  f-4-|  = 


8X3 

7X5 


8 


>  8 


=  4  of 


What  is  the  quotient  of 

45. 

46.  if 

47. 


9 

TV 


48.  H--?- A? 


I  4  9 
T~5  • 

.  ? 

I I  • 

If 


49. 

50. 

51. 

52. 


3* 

54 


16|  -5- 


24? 

3|? 


6|-v-12|? 


3|? 


7  X  5_ 7  X  5 
8  8X3* 


=  3 


53. 

54.  |of^-v-|of  4? 

55.  |L.of 3|-^|of2J? 

56.  ||-i0ffof3|? 


DIVISION  OF  FRACTIONS. 


65 


57.  If  a  family  use  £  of  a  barrel  of  flour  in  a  month, 
how  long  will  2^  barrels  last? 

58.  If  a  bushel  of  corn  cost  $f,  how  many  bushels  can 
be  bought  for  S6-J-?  For  $9f  ? 

59.  If  13  yards  of  silk  cost  S17-J-,  how  many  yards  can 
be  bought  for  $48f  ?  For  $62^-? 

60.  If  a  man  walk  3f%  miles  an  hour,  in  how  many 
hours  will  he  walk  20^  miles? 

61.  At  $33^  an  acre,  how  many  acres  of  land  can  be 
bought  for  $841f  ? 

62.  By  what  must  f  be  multiplied  that  the  product  may 
be  26 J  ? 

63.  Divide  the  product  of  6^-  multiplied  by  3^  by  the 
quotient  of  4^  -r-  5^  ? 

PRINCIPLES  AND  RULES. 

101.  Principles. — 1.  The  quotient  of  two  fractions  having 
a  common  denominator,  equals  the  quotient  of  their  numerators. 

2.  The  multiplying  of  both  dividend  and  divisor  by  the  same 
number  does  not  change  the  value  of  the  quotient. 

102.  Rules. — To  divide  a  fraction  by  a  fraction,  1.  Be- 
duce  the  fractions  to  a  common  denominator,  and  divide  the 
numerator  of  the  dividend  by  the  numerator  of  the  divisor.  Or, 

2.  Invert  the  terms  of  the  divisor,  and  then  multiply  the 
numerators  together  and  also  the  denominators.  Or, 

3.  Multiply  both  dividend  and  divisor  by  the  least  common 
multiple  of  the  denominators  of  the  fractions,  and  divide  the 
resulting  dividend  by  the  resulting  divisor. 

Notes. — 1.  The  third  rule  depends  on  the  second  principle;  and, 
since  multiplying  two  fractions  by  their  least  common  multiple 
changes  them  to  integers,  the  new  dividend  and  divisor  are  always 
integral.  Thus,  multiplying  both  fractions  by  24,  the  l.  c.  m., 
t  iV—  15  -s-  14  =  ;  multiplying  by  6,  the  l.  c.  m.,  6f-r-5|  = 

40  -4-  33  =  1  gtj-.  Compound  fractions  should  first  be  reduced  to  simple 
fractions. 

2.  It  is  not  necessary  that  the  pupil  be  made  equally  familiar  with 
these  three  methods  of  dividing  one  fraction  by  another.  He  should 
thoroughly  master  one  of  them. 

C.Ar. — 6. 


66 


COMPLETE  ARITHMETIC. 


COMPLEX  FRACTIONS. 

4 

64.  Reduce  the  complex  fraction  to  its  simplest  form. 

T 

Process  :  |  =  f  -s-  f  =  =  xf>  ^ws- 

7  0  X  O 

Reduce  to  the  simplest  form 


65. 

3 

8 

69. 

iet 

-4 

CO 

• 

5  of  6 

6  U1  7 

77. 

A  4- 
8 

3 

T¥ 

9 

T¥ 

25 

foi't 

3  _ 

8 

3 

T¥ 

66. 

f 

70. 

25 

74. 

*of2* 

78. 

5 

T 

¥ 

24 

let 

3  _L 

¥ 

67. 

15 

5 

71. 

12* 

5 

75. 

3  of  A 

8  UA  9 

5 

79. 

7 

fx 

¥ 

1 

9 

¥ 

1  2 

4 

6 

68. 

72. 

5 

¥ 

76. 

6, 

_  1  1 

80. 

2 

9. 

3 

¥ 

H 

12-j 

4  n**  1  4 

T  01  ¥¥ 

3  * 
¥ 

5 

9* 

103.  A  complex  fraction  is  simply  an  expressed  division, 
the  numerator  being  the  dividend  and  the  denominator  the 
divisor.  It  is  reduced  to  its  simplest  form  by  performing 
the  division  as  expressed. 

Notes. — 1.  A  complex  fraction  may  be  changed  to  a  fraction  with 
integral  terms,  by  multiplying  both  of  its  terms  by  the  least  common  mul¬ 
tiple  of  the  denominators  of  its  fractions.  (Art.  102,  Note  1.)  Compound 
fractions  must  first  be  reduced  to  simple  fractions. 

2.  Let  the  above  problems  also  be  solved  by  this  method. 


NUMBERS  PARTS  OF  OTHER  NUMBERS. 

MENTAL  PROBLEMS. 

1.  If  ^  of  a  barrel  of  flour  cost  $3,  what  will  a  barrel 
cost  ? 

2.  If  -J-  of  a  ream  of  note  paper  cost  75  cents,  what 
will  a  ream  cost  ? 

3.  Charles  gave  Henry  7  marbles,  which  were  -J-  of  all 
he  had :  how  many  marbles  had  Charles  ? 


FRACTIONS. 


67 


4.  15  is  }  of  what  number? 

5.  16  is  ^  of  what  number? 

6.  12^  is  J  of  what  number? 

7.  16 J  is  y1^  of  what  number? 

8.  22f  is  -J-  of  what  number? 

9.  24  is  f  of  what  number? 

Solution. — If  24  is  f  of  a  number,  ^  is  |  of  24,  which  is  12.  If 
12  is  ^  of  a  number,  f  is  5  times  12,  or  60.  Hence,  24  is  f  of  60. 

10.  27  is  -J  of  what  number? 

11.  45  is  of  what  number? 

12.  64  is  of  what  number? 

13.  27^-  is  -J  of  what  number? 

14.  46f  is  of  what  number? 

15.  37 J-  is  -§  of  what  number? 

16.  87-J-  is  ^  °f  what  number? 

17.  45  is  y  of  how  many  times  9? 

18.  63  is  }  of  how  many  times  12? 

19.  80  is  of  how  many  times  20? 

20.  108  is  }}  of  how  many  times  15? 

21.  What  part  of  4  is  1  ?  What  part  of  4  is  3? 

22.  What  part  of  6  is  5  ?  9  is  8  ?  12  is  6  ? 

23.  11  is  7?  16  is  12?  20  is  15?  18  is  12?  30  is  15? 

24.  7  is  what  part  of  21  ?  8  of  32  ?  9  of  27  ? 

25.  13  of  39?  16  of  72?  15  of  25?  60  of  90? 

26.  \  is  what  part  of  f?  J  of  J?  }  of  }  ?  -J  of  -J? 

97  1  1  ?  2  3  ?  3  5  ?  4  nf  9  ?  5  Af  11? 

L  •  •  Uf  01  ¥  *  ¥  01  ¥  •  ¥  01  ¥  •  T  01  T¥  •  "6  01  T2  * 

28.  |  of  11?  f  of  4?  f-  of  10?  f  of  8?  j-  of  10? 

29.  5i  of  16|?  6|  of  331?  121  of  37|?  33i  of  16|? 

30.  3|  of  6|?  5i  of  2J?  21  of  3i?  6J  of  12}? 

PRINCIPLE  AND  RULE. 

104.  Principle. — Only  like  numbers  can  be  compared. 

105.  Rule. — To  find  what  part  one  number  is  of  another, 
Divide  the  number  denoting  the  part  by  the  number  denoting 
the  whole . 


68 


COMPLETE  ARITHMETIC. 


REVIEW  OF  FRACTIONS. 

MENTAL  PROBLEMS. 

1.  A  boy  having  gave  for  a  knife:  how  much  money 
had  he  left? 

2.  If  ^  be  added  to  a  certain  fraction,  the  sum  will  be  : 
what  is  the  fraction  ? 

3.  A  laborer  spends  -§  of  his  wages  for  board  and  -J  for 
clothing:  what  part  has  he  left? 

4.  A  man  did  ^  of  a  piece  of  work  the  first  day,  \  of  it 
the  second  day,  ^  of  it  the  third  day,  and  the  remainder  the 
fourth  day:  what  part  of  the  work  did  he  do  the  fourth  day? 

5.  A  man  bought  a  farm,  paying  J-  of  the  price  down,  -| 
of  it  the  first  year,  the  second  year,  and  the  remainder 
the  third  year:  what  part  did  he  pay  the  third  year? 

6.  A  man  is  42  years  of  age,  and  -f-  of  his  age  equals  the 
age  of  his  son :  how  old  is  the  son  ? 

7.  A  man  bought  a  cow  for  $33J  and  sold  her  for  of 
what  she  cost :  how  much  did  he  lose  ? 

8.  If  a  yard  of  velvet  cost  $8^,  what  will  f  of  a  yard 
cost  ? 

9.  Jane’s  age  is  16f  years,  and  Mary’s  age  is  |-  of  Jane’s: 
how  old  is  Mary? 

10.  A  man  owning  f  of  a  mill  sells  J-  of  his  share:  what 
part  of  the  mill  does  he  still  own? 

11.  Charles  bought  f  of  a  pound  of  candy  and  gave  his 
sister  f  of  a  pound,  and  his  playmate  J-  of  what  remained: 
what  part  of  a  pound  had  he  left? 

12.  A  wife  is  35  years  of  age,  and  her  age  is  f  of  the  age 
of  her  husband:  how  old  is  her  husband? 

13.  The  difference  between  f  and  f  of  a  certain  number 
is  14:  what  is  the  number? 

14.  A  farmer  sold  50  sheep,  which  were  f  of  his  flock: 
how  many  sheep  had  he  before  the  sale? 

15.  When  Charles  is  -f  older  than  he  now  is,  he  will  be 
21  years  of  age:  how  old  is  he? 


REVIEW  PROBLEMS. 


69 


16.  A  farmer  sold  f  of  his  farm  for  $1645:  at  this  rate, 
what  was  the  value  of  the  farm  ? 

17.  A  man  sold  f  of  his  farm  and  had  64  acres  left: 
how  many  acres  had  he  at  first  ? 

18.  A  man  sold  a  horse  for  $90,  which  was  f  more  than 
it  cost  him:  what  was  the  cost  of  the  horse? 

19.  A  lady  paid  $30  for  a  cloak,  which  was  f  more  than 
she  paid  for  a  dress:  what  was  the  cost  of  the  dress? 

20.  f  of  42  is  yy  of  what  number? 

21.  A  man  is  45  years  old,  and  f-  of  his  age  is  -f-  of  the 
age  of  his  wife:  how  old  is  his  wife? 

22.  Samuel  is  §  as  old  as  Charles,  and  Harry,  who  is  9 
years  old,  is  f  as  old  as  Charles:  how  old  are  Charles  and 
Samuel  ? 

23.  A  man  gave  $150  for  a  watch  and  chain,  and  the 
chain  cost  f  as  much  as  the  watch:  what  did  each  cost? 

24.  If  to  A’s  age  there  be  added  -§  and  f  of  his  age,  the 
sum  will  be  62  years :  what  is  A’s  age  ? 

25.  A  farmer’s  sheep  are  in  4  fields;  the  first  contains  -J 
of  all,  the  second  | ,  the  third  and  the  fourth  52  sheep : 
how  many  sheep  in  the  4  fields? 

26.  A  saddle  cost  $35,  and  f  of  the  cost  of  the  saddle 
was  J-  of  the  cost  of  a  bridle :  wdiat  was  the  cost  of  the 
bridle  ? 

27.  If  to  f  of  a  man’s  age  15  years  be  added,  the  sum 
will  be  f  of  his  age :  how  old  is  he  ? 

28.  The  distance  from  Cleveland  to  Columbus  is  138 
miles,  ff  of  which  is  f  of  the  distance  from  Columbus  to 
Cincinnati :  what  is  the  distance  from  Columbus  to  Cin¬ 
cinnati  ? 

29.  f  is  f  of  what  number? 

30.  If  -|  of  the  value  of  a  house  equal  f-  of  the  value  of 
a  lot,  and  the  value  of  both  is  $4400,  what  is  the  value  of 
each  ? 

31.  If  |  of  A’s  money  equal  f  of  B’s,  and  both  together 
have  $340,  how  much  has  each? 


70 


COMPLETE  ARITHMETIC. 


32.  If  |  of  A’s  age  is  f  of  B’s,  and  -J  of  B’s  is  20  years, 
what  is  the  age  of  each  ? 

33.  If  -f-  of  a  yard  of  velvet  cost  $>2f,  what  will  of  a 
yard  cost? 

34.  How  many  pounds  of  honey,  at  $f  a  pound,  can  be 
bought  for  $3? 

35.  How  many  bushels  of  apples,  at  $f  a  bushel,  can  be 
bought  for  $16§  ? 

36.  If  a  barrel  hold  2f  bushels,  how  many  barrels  will 
be  required  to  pack  55  bushels  of  apples? 

37.  If  lb.  of  sugar  cost  $1,  how  much  will  49 \  lb. 
cost  ? 

38.  If  f  of  a  yard  of  silk  cost  $1^,  how  many  yards  can 
be  bought  for  $10J? 

39.  If  3f  yards  of  cloth  cost  $5^,  what  will  6J  yards 
cost  ? 

40.  If  a  train  of  cars  run  f  of  a  mile  in  If-  minutes,  how 
many  miles  will  it  run  in  15  minutes? 

41.  If  4  pounds  of  coffee  cost  $f,  what  will  7 ^  pounds 
cost  ? 

42.  If  12|  tons  of  hay  will  feed  5  horses  a  year,  how 
many  tons  will  feed  8  horses  a  year? 

43.  If  a  rod  5  feet  long  casts  a  shadow  8-J-  feet  long,  what 
is  the  length  of  a  pole  whose  shadow,  at  the  same  time  of 
day,  is  17^-  feet? 

44.  If  3  men  can  do  a  piece  of  work  in  lOf  days,  how 
long  will  it  take  8  men  to  do  it? 

45.  If  a  barrel  of  flour  will  supply  12  persons  4f  weeks, 
how  long  will  it  supply  7  persons? 

46.  A  can  do  a  job  of  work  in  12  days,  and  B  in  10  days : 
how  long  will  it  take  both  to  do  it? 

47.  A  and  B  can  do  a  certain  w^ork  in  8  days,  and  A  can 
do  it  in  12  days :  in  what  time  can  B  do  it  ? 

48.  A  and  B  can  mow  a  field  in  10  days,  and  A  can  mow 
•J  as  much  as  B:  what  part  of  the  field  can  each  mow  in 
1  day?  How  long  will  it  take  each  to  mow  the  field? 

49.  IIow  is  the  value  of  a  proper  fraction  affected  by 


REVIEW  PROBLEMS. 


71 


adding  the  same  number  to  both  of  its  terms.  By  sub¬ 
tracting  the  same  number?  (Illustrate,  taking  f.) 

50.  How  is  the  value  of  an  improper  fraction,  greater 
than  1 ,  affected  by  adding  the  same  number  to  both  of  its 
terms?  By  subtracting  the  same  number?  (Illustrate.) 

WRITTEN  PROBLEMS. 

51.  Add  |,  |,  of  and  3f 

52.  From  -§  of  If  take  f  of  ff. 

53.  From  the  sum  of  27f  and  20f  take  their  difference. 

54.  Multiply  by  35 ;  35  by  -ff;  by  ff ;  3f  by  2f. 

55.  Divide  ff-  by  32 ;  32  by  ff ;  ff  by  f ;  4f  by  3f. 

56.  #  +  *  =  what?  xi  —  t7^?  fix*? 

57.  Multiply  2045f  by  35;  806  by  84| ;  30f  by  16J-. 

58.  Divide  347f  by  15;  692  by  21f ;  19f  by  16f. 

59.  A  farm  is  divided  into  five  fields,  containing  respect¬ 
ively  21f  A.,  34-J  A.,  45  J  A.,  56f  A.,  and  29f  A.:  how 
many  acres  in  the  farm? 

60.  There  are  30f  sq.  yd.  in  a  square  rod:  how  many 
square  rods  in  786f  sq.  yd.? 

61.  A  man  travels  5f  miles  an  hour:  how  long  will  it 
take  him  to  make  a  journey  of  75|  miles? 

62.  At  $8f  a  ton,  how  many  tons  of  hay  can  be  bought 
for  $108f ? 

63.  If  ^  of  an  acre  of  land  cost  $68,  what  will  12f  acres 
cost  ? 

64.  If  f  of  a  yard  of  velvet  cost  $8f,  how  many  yards  can 
be  bought  for  $196f? 

65.  If  a  number  be  diminished  by  f  of  f-f  of  itself,  the 
remainder  will  be  69 :  what  is  the  number  ? 

66.  A  pedestrian  walked  of  his  journey  the  first  day, 
f  of  it  the  second  day,  and  then  had  24  miles  to  travel : 
how  long  was  the  journey? 

67.  A  man  pays  $350  a  year  for  house  rent,  which  is  ff 
of  his  income :  what  is  his  income  ? 

68.  A  man  bequeathed  to  his  wife  $4860,  which  was  f§ 
of  his  estate :  what  was  the  value  of  the  estate  ? 


72 


COMPLETE  ARITHMETIC. 


69.  A  graded  school  enrolls  208  boys,  and  of  the 
pupils  are  girls :  how  many  pupils  are  enrolled  in  the 

school  ? 

70.  A  man  owning  of  a  ship  sells  f  of  his  share  for 
$3480 :  at  this  rate,  what  is  the  value  of  the  ship  ? 

71.  A  owning  f  of  a  mill,  sold  -§  of  his  share  to  B,  and 
\  of  what  he  then  owned  to  C  for  $460 :  what  was  the  value 
of  the  mill  at  the  rate  of  C’s  purchase  ? 

72.  A  owns  ^  of  a  section  of  land;  B,  ^  of  a  section; 
and  C,  T%  as  much  as  both  A  and  B :  what  part  of  a  sec¬ 
tion  does  C  own? 

73.  A  bought  |  of  a  factory  for  $21840,  and  sold  f  of 
his  share  to  B,  and  J-  of  it  to  C :  what  part  of  the  factory 
did  A  then  own? 

74.  A  and  B  together  own  396  acres  of  land,  and  f  of 
A’s  farm  equals  f  of  B’s :  how  many  acres  does  each  own  ? 

75.  A  stock  of  goods  is  owned  by  three  partners,  A  own¬ 
ing  f ,  B  3^,  and  C  the  remainder ;  the  goods  were  sold  at 
a  profit  of  $6160:  what  was  each  partner’s  share? 

76.  -§  of  a  stock  of  goods  was  destroyed  by  fire,  and  f  of 
the  remainder  was  damaged  by  water,  and  the  uninjured 
goods  were  sold  at  cost  for  $5280:  what  was  the  cost  of  the 
entire  stock  of  goods? 

77.  A  man  paid  §  of  his  money  for  a  farm,  of  what 
remained  for  repairs,  ^  of  what  then  remained  for  stock,  £ 
of  what  then  remained  for  utensils,  and  then  had  left  $650 : 
how  much  money  had  he  at  first? 

78.  A  merchant  tailor  has  67f  yards  of  cloth,  from  which 
he  wishes  to  cut  an  equal  number  of  coats,  pants,  and  vests : 
how  many  of  each  can  he  cut  if  they  contain  3f,  2J,  and 
14  yards  respectively? 

79.  An  estate  was  divided  between  two  brothers  and  a 
sister ;  the  elder  brother  received  •§  of  the  estate,  the  younger 
j^-,  and  the  sister  the  remainder,  which  was  $450  less  than 
the  elder  brother  received:  what  part  of  the  estate  did  the 
sister  receive?  What  was  the  value  of  the  estate?  What 
was  each  brother’s  share? 


DECIMALS. 


73 


SECTION  IS. 

DECIMAL  FRACTIONS. 

NUMERATION  AND  NOTATION. 

1.  If  a  unit  be  divided  into  ten  equal  parts,  what  is  one 
part  called? 

2.  If  a  tenth  of  a  unit  be  divided  into  ten  equal  parts, 
what  is  one  part?  What  is  of  y^? 

3.  If  a  hundredth  of  a  unit  be  divided  into  ten  equal 
parts,  what  is  one  part?  What  is  y1^  of  -yj-g-? 

4.  What  part  of  a  tenth  is  a  hundredth  ?  What  part  of 
a  hundredth  is  a  thousandth? 

5.  How  do  the  fractions  y|y,  and  yg30-y  compare  with 
each  other  in  value?  -j^,  yjy,  and  y&W? 

106.  Since  the  fractional  units,  tenths,  hundredths,  thou¬ 
sandths,  etc.,  decrease  in  value  like  the  orders  of  integers, 
they  can  be  expressed  on  a  scale  of  ten.  This  is  done  by 
extending  the  orders  to  the  right  of  units,  and  calling  the 
first  fractional  order  tenths ,  the  second  hundredths ,  the  third 
thousandths,  etc.,  and  placing  a  period  at  the  left  of  the 
order  of  tenths.  Thus,  -fa  is  written  .5;  y§y  is  written  .05; 
yy50y  is  written  .005,  etc. 

Copy  and  read 


(6) 

(7) 

(8) 

(9) 

(10) 

(11) 

.4 

.03 

.002 

.06 

.07 

.005 

.7 

.05 

.004 

.006 

.004 

.4 

.6 

.08 

.006 

.08 

.8 

.07 

.9 

.09 

.007 

.5 

.09 

.009 

12.  How  many  tenths  and  hundredths  in  .25? 
.63?  .78?  .84?  .69?  .39? 

C.Ar— 7. 


In  .45? 


74 


COMPLETE  ARITHMETIC. 


13.  How  many  tenths,  hundredths,  and  thousandths  in 
.325?  In  .246?  .307?  .405?  .056? 

14.  How  many  tenths,  hundredths,  and  thousandths  in 
.045?  In  .407?  .008?  .065?  .607?  .325? 

15.  How  many  hundredths  in  -j1^?  In  yfy?  .34?  .42? 

16.  How  many  thousandths  in  y§-§7?  In  y^ftnr?  .325? 
.065?  .205?  .008?  .046? 

107.  When  the  right-hand  figure  of  a  decimal  denotes 
hundredths,  the  whole  decimal  denotes  hundredths,  and 
when  the  right-hand  figure  denotes  thousandths,  the  whole 
decimal  denotes  thousandths.  Thus,  .25  is  read  25  hun- 
dreths;  .325  is  read  325  thousandths. 


Copy  and  read 


(17) 

(18) 

(19) 

(20) 

(21) 

.15 

.016 

.245 

.8 

.007 

.42 

.024 

.354 

.63 

.038 

.36 

.045 

.403 

.086 

.462 

.50 

.083 

.587 

.369 

.507 

.06 

.007 

.067 

.504 

.45 

108.  When  fractions  denoting  tenths,  hundredths,  thou- 

sandths,  etc. 

,  are  expressed,  like  integers,  on 

the  decimal 

scale,  they  are  said  to  be  expressed  decimally. 

Express  decimally 

(22) 

(23) 

(24) 

(25) 

(26) 

3 

4 

4  5 

7  5 

1  8 

TIT 

1  OUT 

100T 

ITT 

TTTT 

A 

6 

TOOT 

6  3 

1  0  0  O' 

TTTT 

208 

1  COT 

.  6 

ttt 

1  4 

iooo 

2  15 

lTTT 

T?TT 

355 

lTTT 

8 

TTT 

56 

TTT 

40  7 
lTOT 

TFOT 

4  3 

TTTT 

1  2 

40 

500 

1  0  6 

5 

TOT 

TOT 

lTTT 

TTTT 

lTTT 

27.  What 

is  the  name 

of  the  third  decimal  order?  The 

fourth  ?  The  fifth  ?  The  sixth  ? 

28.  What  does  each  significant  figure  of  .0034  denote? 
Of  .00275  ?  Of  .03405  ?  Of  .000325  ?  Of  .030056  ? 


DECIMAL  FRACTIONS. 


75 


Copy  and  read 


(29) 

(30) 

(31) 

(32) 

.246 

.0635 

.00647 

.0307 

.0246 

.00635 

.000647 

.03007 

.708 

.3464 

.04056 

.030007 

.0708 

.03464 

.004056 

.034005 

.3425 

.32875 

.32453 

.450605 

109.  When  a  decimal  fraction  is  expressed  decimally,  the 


right-hand  figure  is  written 
name  of  the  decimal.  Thus, 

Express  decimally 


(33) 

(34) 

Tihr 

6 

1OO00 

-iwu 

3  3 

i  oxnnr 

8 

40  5 

unnnr 

i  Tnnr 

3042 

1  000  0 

3  a  6 

TT700 

50  0  7 

1  00  01 

Express  decimally 

(37) 

7  tenths ; 

24  hundredths; 
29  thousandths ; 
405  thousandths; 
65  millionths ; 
5064  millionths ; 
40056  millionths. 


in  the  order  indicated  by  the 


3  2  5 

100000  lto 

written  .00325. 

(35) 

(36) 

7  L 

29 

100000 

10  0  OTTUTF 

3  7 

6  0  9 

1  0  OWO 

1  ao  0  ootf 

208 

4  04  5 

1  0  0  0  OlT 

TUoooinr 

3056 

lFOOOTF 

3  3  0  3  3 
TU  0  0  070 

38045 

100000 

2  0  4  0  5  6 
ltfOOOFCT 

(38) 

42  ten-thousandths ; 

506  ten-thousandths ; 

4008  ten-thousandths ; 

65  hundred- thousandths ; 
6007  hundred- thousandths ; 
54008  hundred-thousandths ; 
3004  hundred-thousandths. 


39.  Eighty-five  thousandths. 

40.  Four  hundred  and  seven  thousandths. 

41.  Ninety-five  ten-thousandths. 

42.  Six  hundred  and  forty-four  ten-thousandths. 

43.  Seven  thousand  and_eighty-two  ten-thousandths. 

44.  Fifty-seven  hundred-thousandths. 

45.  Seven  hundred  and  eight  hundred-thousandths. 


76 


COMPLETE  ARITHMETIC. 


46.  Nine  thousand  and  forty-eight  hundred-thousandths. 

47.  Six  hundred  and  four  millionths. 

48.  Seven  thousand  six  hundred  and  forty-three  mill¬ 
ionths. 

49.  Forty  thousand  and  sixty-three  millionths. 

110.  An  integer  and  a  decimal  may  be  written  together 
as  one  number,  as  63%  or  6.5;  253-5-3-  or  25.07.  In  reading 
such  mixed  decimal  numbers,  the  integer  and  the  decimal 
are  connected  by  and.  Thus,  4.5  is  read  4  and  5  tenths. 

50.  Read  45.6;  30.25;  204.045;  84.0307. 

51.  Read  2005.045;  408.00075;  3040.0046;  50060.00705. 

52.  Read  400.045;  500.0063;  7000.0084;  60000.00006. 

Suggestion. — In  such  cases  read  the  integer  as  units;  as,  four 
hundred  units  and  forty-five  thousandths.  The  omission  of  the  word 
units  changes  the  mixed  number'  to  a  pure  decimal. 

53.  Read  5600.0084;  40508.0307;  75000.000605. 

54.  Read  300000.000003;  35000000.000035. 

55.  Write  decimally  56Tfo  5  604Tf -fa  ;  400^°^. 

56.  Write  decimally  207^^;  2560To%°o5oW 

57.  Write  300  units  and  348  millionths. 

DEFINITIONS,  PRINCIPLES,  AND  RULES. 

111.  A  Decimal  Fraction  is  a  fraction  whose  de¬ 
nominator  is  ten  or  a  product  of  tens. 

The  word  decimal  is  derived  from  decern,  a  Latin  word  meaning 
ten.  It  is  applied  to  this  class  of  fractions  because  they  arise  from 
the  division  of  a  unit  into  tenths,  as  tenths,  hundredths,  thou¬ 
sandths,  etc.  Such  a  division  of  a  unit  is  a  decimal  division,  and 
the  resulting  parts  of  the  unit  are  decimal  parts. 

Note. — The  decimal  denominators  are  10,  100,  1000,  etc.  They 
are  powers  of  ten.  (Art.  388.) 

112.  Decimal  fractions  may  be  expressed  in  three  ways: 

1.  By  words;  as,  three  tenths,  twelve  hundredths. 

2.  By  writing  the  denominator  under  the  numerator,  in 
the  form  of  a  common  fraction ;  as,  T3y,  y1^. 


DECIMAL  FRACTIONS. 


77 


3.  By  omitting  the  denominator  and  writing  the  fraction 
in  the  decimal  form,  or  decimally;  as,  .3,  .012.  The  de¬ 
nominator  is  understood. 

Note. — Three  tenths,  r3o,  and  .3,  each  express  the  same  decimal 
fraction,  which  is  the  thing  expressed,  and  not  its  expression.  A 
decimal  fraction  can  be  expressed  orally ,  and  is  so  expressed  when 
read  or  dictated.  When  expressed  in  words,  written  or  oral,  the 
decimal  form  is  not  used.  It  is  an  error  to  teach  that  a  decimal 
fraction  depends  on  the  manner  of  its  expression. 

113.  The  Decimal  Point  is  a  period  placed  at  the 
left  of  the  order  of  tenths,  to  designate  the  decimal  orders. 

114.  A  Complex  Decimal  is  a  decimal  ending  at 
the  right  with  a  common  fraction;  as,  .6|-,  .033J. 

115.  A  Mixed  Decimal  Number  is  an  integer  and 
a  decimal  written  together  as  one  number.  It  is  called 
more  simply  a  Mixed  Decimal. 

The  orders  on  the  left  of  the  decimal  point  are  integral , 
and  those  on  the  right  are  decimal.  The  decimal  orders  are 
called  Decimal  Places. 


116.  The  following  table  gives  the  names  of  a  few  in¬ 
tegral  and  decimal  orders,  and  shows  the  relation  between 
them : 

m 


000000000.  00000000 


Integral  Orders.  Decimal  Orders. 


117.  Principles. — 1.  The  denominator  of  a  decimal  frac¬ 
tion  is  1  with  as  many  ciphers  annexed  as  there  are  decimal 
places  in  the  fraction. 


78 


COMPLETE  ARITHMETIC. 


2.  Ten  units  of  any  decimal  order  equal  one  unit  of  the 
next  order  at  the  left.  Hence, 

3.  The  removal  of  a  decimal  figure  one  place  to  the  right 
divides  its  value  by  10,  and  its  removal  one  place  to  the  left 
multiplies  its  value  by  10. 

4.  The  name  of  a  decimal  is  the  same  as  the  name  of  its 
right-hand  order.  Hence, 

5.  A  decimal  is  read  precisely  as  it  would  be  were  the  denom¬ 
inator  expressed. 

118.  Rules. — 1.  To  read  a  decimal,  Read  it  as  though  it 
were  an  integer ,  and  add  the  name  of  the  right-hand  order. 

2.  To  write  a  decimal,  Write  it  as  an  integer ,  and  so  plac,e 
the  decimal  point  that  the  right-hand  figure  shall  stand  in  the 
order  denoted  by  the  name  of  the  decimal. 

Note. — When  the  number  does  not  fill  all  the  decimal  places, 
supply  the  deficiency  by  prefixing  decimal  ciphers. 

WRITTEN  PROBLEMS. 

Express  decimally 

58.  Two  hundred  five  ten-thousandths. 

59.  Forty  thousand  thirty-four  millionths. 

60.  Two  thousand  four  hundred-thousandths. 

61.  Six  hundred  fifteen  ten-millionths. 

62.  Six  hundred  units  and  fifteen  ten- thousandths. 

63.  Fifteen  units  and  fifteen  thousandths. 

64.  Three  hundred  thousand  three  hundred  thirteen  hun¬ 
dred-millionths. 

65.  Five  million  eighty-five  ten-millionths. 

66.  Twelve  hundred-thousandths. 

67.  Four  hundred  units  and  four  hundred  and  sixty-five 

millionths.  / - 

68.  Twenty-five  units  and  twenty-five  thousandths. 

69.  Five  thousand  units  and  five  thousandths. 

70.  Three  hundred  and  seventy-five  units  and  three  hun¬ 
dred  and  seventy-five  billionths. 


REDUCTION  OF  DECIMALS. 


79 


71.  Thirty  thousand  forty-six  hundred-thousandths. 

72.  One  million  forty -five  billionths. 

73.  Eighty  thousand  and  forty  units  and  three  hundred 
and  six  ten-thousandths. 

74.  Fifteen  thousand  units  and  fifteen  ten-thousandths. 

75.  Seventy-five  units  and  five  thousand  and  forty-three 
millionths. 

76.  One  million  units  and  one  millionth. 

REDUCTION  OF  DECIMALS. 

Case  I. 

Decimals  reduced  to  Dower  or  Higher  Orders. 

1.  How  many  tenths  in  6  units?  In  15  units?  In  24 
units  ? 

2.  How  many  hundredths  in  5  tenths  ?  In  .6?  .8?  .7? 

3.  How  many  thousandths  in  .06?  In  .24?  .47?  .55? 

4.  How  many  tenths  in  .60?  In  .70?  .90?  .600?  .700? 
.800?  .5000?  1.50? 

5.  How  many  hundredths  in  .240?  In  .420?  .560? 
.4500?  .8500?  .35000?  .0700? 

WRITTEN  PROBLEMS. 

6.  Reduce  .875  to  millionths. 

Process  :  .875  =  .875000 

7.  Reduce  .0674  to  ten-millionths. 

8.  Reduce  .075  to  hundred-thousandths. 

9.  Reduce  62.7  to  thousandths. 

10.  Reduce  5.33  to  ten-thousandths. 

11.  Reduce  3.  to  hundredths. 

12.  Reduce  45.  to  ten-thousandths. 

13.  Reduce  .04500  to  thousandths. 

Process  :  .04500  =  .045 

14.  Reduce  5.24000  to  hundredths. 


80 


COMPLETE  ARITHMETIC. 


119.  Principles. — 1.  Annexing  ciphers  to  a  decimal  frac¬ 
tion  multiplies  both  of  its  terms  by  the  same  number ,  and  hence 
does  not  change  its  value.  (Art.  85.) 

2.  Cutting  off  ciphers  from  the  right  of  a  decimal  fractipxk. 
divides  both  of  its  terms  by  the  same  number,  and  hence  does  not 
change  its  value.  (Art.  81.) 

Note. — The  annexing  of  decimal  ciphers  to  an  integer  does  not 
change  its  value.  Thus,  12.  =  12.0,  or  12.00;  that  is,  12  units  =  120 
tenths  =  1200  hundredths,  etc. 


Case  II. 

Decimals  reduced  to  Common  Fractions. 

15.  How  many  fifths  in  T\?  -j%?  .2?  .8? 

16.  How  many  fourths  in  y2^-?  -j^nr  •  *25?  .50? 

.75? 

17.  How  many  twentieths  in  -jVf?  iVf?  *20?  -25?  .55? 
.75?  .95? 


WRITTEN  PROBLEMS. 


18.  Reduce  .625  to  a  common  fraction  in  its  lowest  terms. 


Process:  .625  =  x%2^  =  =  f,  Ans. 

Reduce  to  common  fractions  in  lowest  terms 


19.  .125 

20.  .75 

21.  .075 

22.  .0625 

23.  .1625 

24.  .2250 


25.  .004 

26.  .5625 

27.  .0125 

28.  .3525 

29.  3.525 

30.  37.75 


31.  62.025 

32.  37.625 

33.  56.371 

34.  247.331 

35.  16.66f 

36.  214.00^ 


120.  Rule. — To  reduce  a  decimal  to  a  common  fraction, 
Omit  the  decimal  point  and  supply  the  denominator,  and  then 
reduce  the  fraction  to  its  lowest  terms. 


Note. — When  the  denominator  is  written  the  fraction  is  both  deci¬ 
mal  and  common. 


REDUCTION  OF  DECIMALS. 


81 


Case  III. 

Common.  Fractions  reduced  to  Decimals. 

37.  How  many  tenths  in  -J?  In  f?  f? 

38.  How  many  hundredths  in  f?  f?  f? 

39.  How  many  hundredths  in  -^g-?  A?  A? 

40.  How  many  hundredths  in  A?  A?  A?  A? 

WRITTEN  PROBLEMS. 

41.  Change  yfy  to  a  decimal. 

Process.  Since  — tI?  °f  3,  and  since  3  =  3.000 

125  )  3.00  ( .024,  Ans.  (Art.  119,  Note),  T\z  of  3  =  As  of  3.000  = 
2  50  .024.  Or,  ylx  — tts  of  3  units,  and  3  units 

500  =  3000  thousandths,  and  of  3000  tliou- 

500  sandths  =  24  thousandths  =  .024. 


Reduce  to  decimal  fractions 


42. 

5 

"S' 

4^ 

00 

• 

32 

T5 

54. 

1  3 

TO 

60. 

i^A 

43. 

9 

T6- 

49. 

8  7 

ft 

55. 

7 

ToTT 

61. 

25yf 

44. 

3 

To 

50. 

1  2 

1  2T 

56. 

2  3 

2lT(7 

62. 

Q71  3 

^'t<7 

45. 

2  5 

3  2 

51. 

A 

57. 

4 

T2T0 

63. 

5 

irtr  o 

4G. 

64 

T2  5" 

52. 

A 

58. 

1 

Tiro 

64. 

23 

3  017 

47. 

80 

TT5 

53. 

1  9 

M 

59. 

2  1 

4817 

65. 

1  4 

1 1  1 

121.  Rule. — To  reduce  a  common  fraction  to  a  decimal, 
Annex  decimal  ciphers  to  the  numerator  and  divide  by  the 
denominator,  and  point  off  as  many  decimal  places  in  the  quo¬ 
tient  as  there  are  annexed  ciphers. 

Notes. — 1.  When  a  sufficient  number  of  decimal  places  is  obtained, 
the  remainder  may  be  discarded,  or  the  quotient  may  be  expressed 
as  a  mixed  decimal. 

2.  When  the  denominator  of  a  common  fraction  in  its  lowest 
terms  contains  other  prime  factors  than  2  and  5,  the  process  will  not 
terminate. 

3.  When  the  quotient  repeats  the  same  figure,  or  the  same  set  of 
figures,  as  in  problems  63,  64,  and  65,  it  is  called  a  Repeating  Decimal , 
or  a  Circulating  Decimal,  and  the  figure  or  figures  repeated  are  called 
a  Rcpetend.  (Art.  431.) 


82 


COMPLETE  ARITHMETIC. 


ADDITION  OF  DECIMALS. 

1.  Add  16.25,  48.037,  90.0033,  and  .864. 

Since  only  like  orders  can  be  added  (Art.  27), 
write  the  figures  of  the  same  order  in  the  same 
column.  Since  ten  units  of  any  order  equal  one 
unit  of  the  next  higher  order,  begin  at  the  right 
and  add  as  in  simple  numbers.  Place  the  decimal 
point  at  the  left  of  the  1  tenth. 

2.  Add  .375,  80.06,  45.0084,  .00755,  and  84.635. 

3.  Add  84.08,  16.075,  2.9,  1.96,  1.003,  and  5.0008. 

4.  Add  $15.34,  $65,048,  $9,083,  $12.,  $16.66},  $18.06, 
$95.37|,  and  $35.75. 

5.  Add  26.371,  19.081,  23.042},  38.5,  6.00},  and  7^. 

6.  Add  256  thousandths,  3005  millionths,  207  ten-thou¬ 
sandths,  34  ten-millionths,  and  94  hundred-thousandths. 

7.  Add  fifteen  thousandths,  eighty-one  ten-thousandths, 
fifty-six  millionths,  seventeen  ten-millionths,  and  two  hun¬ 
dred  and  five  hundred-thousandths. 

8.  How  many  rods  of  fence  will  inclose  a  field,  the  four 
sides  of  which  are  respectively  46.6  rd.,  50.65  rd.,  24.33} 
rd.,  and  27  rd.  ? 

9.  Five  bars  of  silver  weigh  respectively  .75  lb.,  1.15  lb., 
.86}  lb.,  1.34  lb.,  and  .9  lb.:  what  is  their  total  weight? 

10.  The  average  amount  of  rain  in  San  Francisco  in  the 
winter  months  is  11.25  inches;  in  the  spring,  8.81  inches; 
in  the  summer,  .03  inches;  and  in  the  autumn,  2.75  inches: 
what  is  the  amount  for  the  year? 

122.  Rule. — To  add  decimals,  1.  Write  the  numbers  so  that 
figures  of  the  same  order  shall  stand  in  the  same  column. 

2.  Add  as  in  the  addition  of  integers,  and  place  the  decimal 
point  at  the  left  of  the  tenths’  order  in  the  amount. 

Note. — If  a  mixed  decimal  does  not  contain  as  many  decimal 
places  as  either  of  the  other  numbers,  change  the  terminal  common 
fraction  to  a  decimal,  and  continue  the  division  until  the  requisite 
number  of  decimal  places  is  secured. 


Process. 

16.25 

48.037 

90.0033 

•864 

155.1543,  Ans. 


SUBTRACTION  OF  DECIMALS. 


83 


SUBTRACTION  OF  DECIMALS. 


1.  From  47.625  take  28.7. 


1st  Process. 

47.625 

28.700 

18.925 


2d  Process. 

47.625 

28.7 

18.925 


Reduce  the  decimals  to  a  like 
order  (Art.  119),  and  since  units 
can  only  be  taken  from  like  units, 
write  the  numbers  so  that  figures 


of  the  same  order  shall  stancTTn 


2.  From  46.7  take  29.825.  the  same  column;  and  since  ten 


units  of  any  decimal  order  equal 
one  unit  of  the  next  higher  order, 
subtract  as  in  simple  numbers. 
Place  the  decimal  point  at  the 
left  of  the  tenths’  order. 


2d  Process. 

46.7 

29.825 

16.875 


1st  Process. 

46.700 

29.825 

16.875 


Note. — A  comparison  of  the  two  processes  shows  that  it  is  unnec¬ 
essary  to  fill  the  vacant  orders  with  ciphers. 

3.  From  4.05  take  2.0075. 

4.  From  .6J  take  .0087^-. 

5.  From  12.  take  .0005. 

6.  From  six  tenths  take  six  thousandths. 

7.  From  forty-four  thousandths  take  forty-four  millionths. 

8.  From  301  ten-thousandths  take  4005  millionths. 

9.  From  50065  ten-millionths  take  1307  billionths. 

10.  A  man  walked  33.7  miles  the  first  day  and  28.75 
miles  the  second:  how  much  farther  did  he  walk  the  first 
day  than  the  second  ? 

11.  The  average  amount  of  rain  at  Cincinnati  in  the 
summer  months  is  13.7  inches,  and  in  the  winter  months  it 
is  11.15  inches:  what  is  the  difference? 

12.  The  mean  height  of  the  barometer  at  Boston  is 
29.934  inches,  and  at  Pekin  it  is  30.154  inches:  what  is 
the  difference? 

123.  Rule. — To  subtract  decimals,  1.  Write  the  numbers  so 
that  figures  of  the  same  order  shall  stand  in  the  same  column. 

2.  Subtract  as  in  the  subtraction  of  integers ,  and  place  the 
decimal  point  at  the  left  of  the  tenths'  order  in  the  remainder. 


84 


COMPLETE  ARITHMETIC. 


MULTIPLICATION  OF  DECIMALS. 


1.  How  much  is  7  times  T*g?  7  times  T4g?  8  times 

2.  How  much  is  8  times  yj-g  ?  8  times  y^-g  ?  6  times  yf-g  ? 

3.  What  is  the  product  of  ^  X  TV  txt  X  A?  t8tt  X  TV 

4.  What  is  the  product  of  y1^  X  -^g-g?  t4o  X  rgir? 

5.  What  is  the  product  Of  -y^-g  by  yyg?  yjg  by  yf-g? 

6.  What  is  the  denominator  of  the  product  when  tenths 
are  multiplied  by  units?  Tenths  by  tenths?  Tenths  by 
hundredths  ?  Hundredths  by  hundredths  ?  Hundredths  by 
thousandths  ? 

7.  What  is  the  denominator  of  the  product  of  any  two 
fractions  whose  denominators  are  powers  of  10? 


WRITTEN  PROBLEMS. 


8.  Multiply  .625  by  .23. 


Process. 

.625 

.23 

1875 

1250 

.14375 


Since  .625  =  Amy,  and  .23  =  T2g3g,  .625  X  -23  =  T6^2o5g 
X  T2o3g  =  tWoVV  =  .14375.  Hence,  .625  X  .23  =  .14375. 
Since  thousandths  multiplied  by  hundredths  produce  hun¬ 
dred-thousandths,  the  product  contains  Jive  decimal  places, 
or  as  many  as  both  of  the  factors. 


Multiply 


9. 

6.5 

by 

.75 

14. 

4.36 

by  .27 

19. 

.085 

by 

• 

O 

CO 

-  10. 

.043 

by 

6.5 

15. 

64. 

by  .032 

20. 

2.56 

by 

250. 

11. 

.0432 

by 

5.4 

16. 

30.3 

by  .018 

21. 

3.24 

by 

.33^ 

12. 

.048 

by 

24. 

17. 

.056 

by  24. 

22. 

5.75 

by 

83 

13. 

5.6 

by 

.056 

18. 

50. 

by  .08 

23. 

16J 

by 

.045 

24.  Multiply  sixteen  thousand  by  sixteen  thousandths. 

25.  Multiply  205  millionths  by  46  thousandths. 

26.  Multiply  6.25  by  10.  By  100. 


Since  the  removal  of  a  decimal  figure  one 

place  to  the  left  multiplies  its  value  by  10 

(Art.  117,  Pr.  3),  the  removal  of  the  decimal 

point  one  place  to  the  right  multiplies  6.25  by 

10,  and  the  removal  of  the  point  two  places  to  the  right  multiplies 

6.25  bv  100. 

0 


Process. 

6.25X  10  =62.5 
6.25  X  100  =  625. 


DIVISION  OF  DECIMALS. 


85 


27.  Multiply  3.406  by  100.  By  1000. 

28.  Multiply  .00048  by  1000.  By  100000. 

29.  Multiply  .0000256  by  10000.  By  1000000. 

PRINCIPLES  AND  RULES. 

124.  Principles. — 1.  The  number  of  decimal  places  in  the 
product  equals  the  number  of  decimal  places  in  both  factors. 

2.  Each  removal  of  the  decimal  point  one  place  to  the  right, 
multiplies  the  decimal  by  10. 

125.  Rules. — 1.  To  multiply  one  decimal  by  another, 
Multiply  as  in  the  multiplication  of  integers,  and  point  off  as 
many  decimal  places  in  the  product  as  there  are  decimal  places 
in  both  multiplicand  and  multiplier. 

Note. — If  there  be  not  enough  decimal  figures  in  the  product, 
supply  the  deficiency  by  prefixing  decimal  ciphers. 

2.  To  multiply  a  decimal  by  10,  100,  1000,  etc.,  j Remove 
the  decimal  point  as  many  places  to  the  right  as  there  are  ciphers 
in  the  multiplier. 

Note. — If  there  be  not  enough  decimal  places  in  the  product, 
supply  the  deficiency  by  annexing  ciphers. 


DIVISION  OF  DECIMALS. 


1.  How  many  times  are  5  tenths  contained  in  10  tenths? 
7  tenths  in  35  tenths? 

2.  How  many  times  are  7  hundredths  contained  in  21 
hundreths?  7  hundredths  in  35  hundredths? 

Q  Whnt  i~  9.39  27.  99  7 

o.  u  nai  is  Tir  —  TTr  r  too- ~  uni  •  To' 

4.  What  is  .8-1-. 4?  .21--. 07?  .084--. 012? 

5.  What  is  --  yfy  ?  toW  ? 

Suggestion. — Reduce  the  fractions  to  a  common  denominator. 


.  25  9 

•  1  0  0  0  • 


15  ? 

nnnr* 


6.  What  is  .3 -4-. 15?  .25--. 125?  .12-- .012? 

7.  Of  what  order  is  the  quotient  when  tenths  are  divided 
by  tenths?  Hundredths  by  hundredths?  Thousandths  by 
thousandths  ? 


86 


COMPLETE  ARITHMETIC. 


8.  Of  what  order  is  the  quotient  when  any  order  is 
divided  by  a  like  order  ?  When  any  number  is  divided 
by  a  like  number? 

WRITTEN  PROBLEMS. 

9.  Divide  8.05  by  .35 

Process. 

.35  )  8.05  (  23.,  Ans.  35  hundredths  are  contained  in  805  hun- 

7  0  dredths,  a  like  number,  23  times,  and  hence 

1  05  8.05  — ; —  .35  =  23.  The  quotient  is  units. 

105 


10.  Divide  80.5  by  .35 

Process.  By  annexing  a  decimal  cipher  to  80.5, 

.35  )  30.50  (  230.,  Ans.  w}1jc}1  does  not  change  its  value  (Art.  119), 

the  dividend  and  divisor  are  made  like  num- 


10  5 
10  5 


0 


bers,  and  hence  their  quotient  is  units.  80.50 
.35  =  230. 


11.  Divide  .805  by  .35 

Process. 


.35  )  .805  (  2.3,  Ans. 
70_ 

105 

105 


Since  .35  and  .80,  the  first  partial  dividend, 
are  like  numbers,  the  first  quotient  figure  (2) 
denotes  units ;  and  if  the  first  figure  denotes 
units,  the  second  must  denote  tenths.  Hence. 
.805  -4-  .35  =  2.3 


The  pointing  in  all  the  cases  in  the  division  of  decimals,  may  also 
be  explained  on  the  principle,  that  the  dividend  is  the  product  of  the 
divisor  and  quotient,  and  hence  it  must  contain  as  many  decimal  places 
as  both  divisor  and  quotient. 

In  the  9th  example,  the  divisor  and  dividend  contain  an  equal 
number  of  decimal  places,  and  hence  there  are  no  decimal  places  in 
the  quotient. 

In  the  10th  example,  the  divisor  contains  one  more  decimal  place 
than  the  dividend,  and  hence  a  decimal  place  must  be  added  to  the 
dividend  before  the  division  is  possible. 

In  the  11th  example,  the  divisor  contains  two  decimal  places 
and  the  dividend  three ,  and  hence  the  quotient  contains  one  decimal 
place. 


DIVISION  OF  DECIMALS. 


8T 


Divide 

12. 

32.4  by  1.8 

25. 

6.241  by  .0079 

13. 

2.56  by  .64 

26. 

67.5  by  .075 

14. 

.288  by  .036 

27. 

.675  by  75. 

15. 

82.5  by  2.75 

28. 

6.75  by  750. 

16. 

62.5  by  .025 

29. 

256.  by  .075 

17. 

9.  by  .45 

30. 

.256  by  250. 

18. 

4.53  by  .0302 

31. 

.0025  by  50. 

19. 

.3  by  .0125 

32. 

25.  by  .00125 

20. 

.625  by  12.5 

33. 

.001  by  100. 

21. 

.0256  by  .32 

34. 

100.  by  .001 

22. 

17.595  by  8.5 

35. 

.045  by  900. 

23. 

3.3615  by  12.45 

36. 

$13.50  by  $.37|- 

24. 

.031812  by  4.82 

37. 

$12.  by  $.06J 

38.  Divide  twenty-four  thousandths  by  sixteen  millionths. 

39.  Divide  seventy-eight  by  thirty-four  thousandths. 

40.  Divide  fifteen  millionths  by  six  hundredths. 

41.  Divide  45.7  by  10.  By  100. 

Process.  Since  the  removal  of  a  decimal  figure  one 

^  y  ^  gy  place  to  the  right  divides  its  value  by  10 

45  7  ioo _  457  (Art.  117,  Pr.  3),  the  removal  of  the  decimal 

point  one  place  to  the  left  divides  a  decimal  by 
10,  and  the  removal  of  the  point  two  places  to  the  left  divides  it 
by  100. 

42.  Divide  483.75  by  100.  By  1000. 

43.  Divide  54.50  by  100.  By  10000. 

44.  Divide  .005  by  1000.  By  100. 


PRINCIPLES  AND  RULES. 

126.  Principles. — 1.  Since  the  dividend  is  the  product 
of  the  divisor  and  quotient ,  it  contains  as  many  decimal  places 
as  both  divisor  and  quotient.  Hence, 

2.  The  quotient  must  contain  as  many  decimal  places  as  the 
number  of  decimal  places  in  the  dividend  exceeds  the  number  of 
decimal  places  in  the  divisor.  Hence, 


88 


COMPLETE  ARITHMETIC. 


3.  When  the  divisor  and  dividend  contain  the  same  number 
of  decimal  places,  the  quotient  is  units. 

4.  The  dividend  must  contain  as  many  decimal  places  as  the 
divisor  before  division  is  possible. 

5.  Each  removal  of  the  decimal  point  one  place  to  the  left 
divides  a  decimal  by  10. 

127.  Rules. — 1.  To  divide  one  decimal  by  another,  Divide 
as  in  the  division  of  integers,  and  point  off  as  many  decimal 
places  in  the  quotient  as  the  number  of  decimal  places  in  the 
dividend  exceeds  the  number  in  the  divisor. 

Rotes. — 1.  When  the  divisor  contains  more  decimal  places  than 
the  dividend,  supply  the  deficiency  in  the  dividend  by  annexing  deci¬ 
mal  ciphers. 

2.  When  the  quotient  has  not  enough  decimal  figures,  supply  the 
deficiency  by  'prefixing  decimal  ciphers. 

3.  When  there  is  a  remainder,  the  division  may  be  continued  by 
annexing  ciphers,  each  cipher  thus  annexed  adding  one  decimal  place 
to  the  dividend.  Sufficient  accuracy  is  usually  secured  by  carrying 
the  division  to  four  or  five  decimal  places. 

2.  To  divide  a  decimal  by  10,  100,  1000,  etc.,  Remove  the 
decimal  point  as  many  places  to  the  left  as  there  are  ciphers  in 
the  divisor. 

REVIEW  PROBLEMS. 

1.  Reduce  yf-g-  to  a  decimal. 

2.  Reduce  -25W  to  a  decimal. 

3.  Change  .325  to  a  common  fraction. 

4.  Change  .0045  to  a  common  fraction. 

5.  From  the  sum  of  67.5  and  .54  take  their  difference. 

6.  From  the  sum  of  64.5  and  .015  take  their  product. 

7.  Multiply  6.25  +  .075  by  6.25  — .075. 

8.  Divide  .0512  by  .032  X  .005. 

9.  From  25.6  .064  take  32.4  X  .015. 

10.  What  is  the  value  of  $5.33  X  2.5  -j-.075? 

*11.  What  is  .08J  x  1.2i -r- .0061  X  .016? 

12.  Multiply  15  millionths  by  7  million. 

13.  Divide  16  ten-millionths  by  25  thousandths. 

14.  Divide  205  millions  by  41  ten-thousandths. 

s;:See  “Note”  on  page  386 


UNITED  STATES  MONEY. 


89 


SECTION  X. 


UNITED  STATES  MONET. 


PRELIMINARY  DEFINITIONS. 

128.  United  States 
31oney  is  the  legal  cur¬ 
rency  of  the  United  States. 

It  is  also  called  Federal 


129.  The  denominations 
used  in  business  and  ac¬ 
counts,  are  dollars ,  cents,  and 
mills.  A  dollar  equals  100 
cents,  and  a  cent  equals  10 
mills. 

The  figures  denoting  dol¬ 
lars  are  separated  from  those  denoting  cents  by  a  decimal 
point,  called  a  Separatrix,  and  they  are  preceded  by  the 
character,  8,  called  the  Dollar  Sign. 

130.  The  first  two  figures  at  the  right  of  dollars  denote 
cents,  and  the  third  figure  denotes  mills.  The  two  figures 
denoting  cents  express  hundredths  of  a  dollar,  and  the  figure 
denoting  mills  expresses  tenths  of  a  cent,  or  thousandths  of  a 
dollar.  The  three  figures  denoting  cents  and  mills  may  be 
read  together  as  so  many  thousandths  of  a  dollar. 

Notes. — 1.  United  States  Money  consists  of  Coin  and  Paper  Money. 
Coin  is  called  Specie  Currency  or  Specie,  and  paper  money  is  called 
Paper  Currency. 

2.  The  principal  gold  coins  are  the  double  eagle  ($20),  eagle  ($10), 
half-eagle,  quarter-eagle,  three-dollar  piece,  and  dollar. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dollar,  and 
dime.  The  smaller  coins  are  the  five-cent  piece,  three-cent  piece, 
two-cent  piece,  and  cent,  the  first  two  being  made  of  copper  and 
nickel,  and  the  last  two  of  bronze,  an  alloy  of  copper,  tin,  and  zinc. 

C.  Ar.— 8. 


90 


COMPLETE  ARITHMETIC. 


3.  Gold  and  silver  coins  are  alloyed,  to  make  them  harder  and 
more  durable.  The  gold  coins  contain  9  parts  of  gold  and  1  part  of 
copper;  and  the  silver  coins  contain  9  parts  of  silver  and  1  part  of 
copper.  Nickel  and  copper  coins  are  made  in  the  proportion  of  1 
part  nickel  to  3  parts  copper. 

4.  Paper  money  consists  of  notes  issued  by  the  United  States, 
called  Treasury  Notes,  and  bank  notes  issued  by  banks. 


131.  NOTATION  AND  REDUCTION. 

1.  Express  in  words,  $75.50;  $105.08;  $1000.45;  $15080.; 
$.87;  $.375;  $5. 

2.  Express  in  words,  $37,507;  $250,075;  $80,005;  $.075; 
$2080.375;  $100,058;  $.065. 

3.  Read  decimally,  $70.25;  $140.05;  $387.60;  $560.09; 
$84.37;  $.08. 

4.  Read  decimally,  $.255;  $16,455;  $300,056;  $475,005; 
$1005.375;  $240,061;  $.005. 

WRITTEN  PROBLEMS. 

5.  Write,  in  figures,  ten  dollars  fifty  cents. 

6.  Write  forty  dollars  sixty  cents  five  mills. 

7.  Write  100  dollars  37  cents  4  mills. 

8.  Write  430  dollars  5  cents;  25  dollars  5  mills. 

9.  Write  75  cents  6  mills;  6  cents  5  mills. 

10.  Write  10  mills;  10  cents  4  mills. 

11.  How  many  cents  in  $25?  $100?  $350? 

12.  How  many  mills  in  $47  ?  $150  ?  $165  ? 

13.  How  many  mills  in  $.75?  $.625?  $.017? 

14.  How  many  cents  in  $5.37?  $16.85?  $40.08? 

15.  How  many  mills  in  $.37^?  $4.62J?  $10? 

16.  Reduce  1500  cents  to  dollars. 

17.  Reduce  15000  mills  to  dollars. 

18.  Reduce  450  mills  to  cents. 

19.  Reduce  $25.08  to  mills. 

20.  Reduce  $100.01  to  cents;  to  mills. 


UNITED  STATES  MONEY. 


91 


ADDITION  AND  SUBTRACTION. 

1.  A  man  paid  $7.50  for  a  pair  of  boots,  and  $5.50  for 
a  hat:  how  much  did  he  pay  for  both? 

2.  A  lady  paid  $15  for  a  shawl,  $5.75  for  a  hat,  $2.25 
for  a  pair  of  gloves,  and  $4  for  a  pair  of  gaiters :  what  was 
the  amount  of  her  purchases? 

3.  A  drover  bought  cows  at  $36.50  a  head,  and  sold 
them  at  $40  a  head :  how  much  did  he  gain  ? 

4.  A  man  bought  a  coat  for  $24.25,  and  a  vest  for  $4.50, 
and  handed  the  merchant  three  $10  bills :  how  much  money 
did  he  receive  back? 

5.  A  mechanic  earns  $20  a  week,  and  his  family  expenses 
amount  to  $16.75  a  week:  how  much  has  he  left? 

6.  A  bookseller  bought  a  set  of  maps  for  $17,  and  a  set 
of  charts  for  $6.50,  and  sold  both  sets  for  $28.50:  how  much 
did  he  gain  ? 

WRITTEN  PROBLEMS. 

7.  What  is  the  sum  of  $.65,  $15.44,  $60.62J,  $100, 
$94.05,  and  $.87J? 

8.  From  $100.15  take  $62,371 

9.  To  the  sum  of  $308.60  and  $190,125  add  their  dif¬ 
ference. 

10.  From  the  sum  of  $2750.  and  $1680.62^-  take  their 
difference. 

11.  A  merchant’s  sales  for  a  week  were  as  follows: 
Monday,  $125.60;  Tuesday,  $98.50;  Wednesday,  $190.30; 
Thursday,  $215.;  Friday,  $175.80;  Saturday,  $247.90:  what 
was  the  amount  of  his  sales  for  the  week? 

12.  A  man  exchanged  three  city  lots,  valued  respectively 
at  $900,  $1200,  and  $750,  for  a  farm  valued  at  $3075,  pay¬ 
ing  the  difference  in  money :  how  much  money  did  he  pay  ? 

13.  A  man  receiving  a  salary  of  $1600  a  year,  pays  $325 
for  house  rent,  $450.80  for  provisions,  $200.60  for  clothing, 
and  $245  for  all  other  expenses:  how  much  has  he  left? 


92 


COMPLETE  ARITHMETIC. 


14.  A  man  deposits  in  a  bank,  at  different  times,  $75, 
$230.80,  $180.40,  and  $95,  and  he  draws  out  $40,  $87.50, 
$331.45,  $20.15,  and  $18.60:  what  is  his  hank  balance? 

132.  Rule. — To  add  or  subtract  sums  of  money,  Write 
units  of  the  same  denomination,  in  the  same  column ,  add  or  sub¬ 
tract  as  in  simple  numbers ,  and  separate  dollars  and  cents  by  a 
decimal  point,  and  prefix  the  dollar  sign. 

MULTIPLICATION  AND  DIVISION. 

1.  A  mechanic  earns  $2.50  a  day:  how  much  will  he 
earn  in  6  days?  10  days? 

2.  What  will  8  barrels  of  flour  cost,  at  $7.25  a  barrel? 
At  $6.50  a  barrel? 

3.  What  will  20  yards  of  carpeting  cost,  at  $1.75  a  yard? 
At  $2.25  a  yard? 

4.  A  drover  paid  $38.70  for  9  sheep:  what  did  they  cost 
apiece  ? 

5.  A  man  paid  $42  for  8  tons  of  coal:  what  did  it  cost 
per  ton? 

6.  If  a  man  earn  $39  in  6  days :  how  much  will  he  earn 
in  10  days?  In  20  days? 

7.  At  25  cents  a  dozen,  how  many  dozens  of  eggs  can 
be  bought  for  $4.50? 

WRITTEN  PROBLEMS. 

8.  A  farmer  sold  45  hogs  at  $22.45  apiece:  how  much 
did  he  receive  for  them? 

9.  A  miller  sold  237  pounds  of  flour,  at  $7.62-|-  a  barrel : 
what  amount  did  he  receive? 

10.  A  man  sold  a  farm  of  260  acres,  at  $33^-  per  acre: 
what  was  the  amount  received? 

11.  A  farm  containing  125  acres  was  sold  for  $5093.75: 
what  was  the  price  per  acre? 

12.  How  many  carriages,  at  $125  apiece,  can  be  bought 
for  $8000  ?  For  $7500  ? 


LEDGER  COLUMNS. 


93 


13.  At  $12. 37^  a  ton,  how  many  tons  of  hay  can  be 
bought  for  $4653?  For  $1163.25? 

14.  A  farmer  sold  3  hogs,  weighing  respectively  278, 
309,  and  327  pounds,  at  $.07^-  a  pound:  how  much  did  he 
receive  ? 

15.  A  farmer  sold  in  one  year  536  pounds  of  butter,  at 
30  cts.  a  pound;  1200  pounds  of  cheese,  at  16-J  cts. ;  and  19 
tons  of  hay,  at  $8.75  a  ton :  how  much  did  he  receive? 

16.  A  grocer  bought  540  pounds  of  coffee  for  $81,  and 
420  pounds  of  tea  for  $525 ;  he  sold  the  coffee  at  18  cts.  a 
pound,  and  the  tea  at  $1.60  a  pound :  how  much  did  he  gain  ? 

133.  Rules. — 1.  To  multiply  or  divide  sums  of  money  by 
an  abstract  number,  Multiply  or  divide  as  in  simple  numbers, 
separate  dollars  and  cents  in  the  result  by  a  decimal  'point,  and 
prefix  the  dollar  sign. 

2.  To  divide  one  sum  of  money  by  another,  Reduce  both 
numbers  to  the  same  denomination ,  and  divide  as  in  simple 
numbers. 


ABBREVIATED  METHODS. 


LEDGER  COLUMNS. 


134.  A  Ledger  is  a  book 
in  which  business  men  keep 
a  summary  of  accounts. 

The  items  on  a  ledger  page 
often  make  long  columns  o 
figures,  which  are 
footed  with  absolute  accu¬ 
racy. 

135.  Let  the  pupil  foot  the 
following  ledger  columns  by 
adding  two  or  more  columns 
at  once,  being  as  careful  to 
obtain  accurate  results  as  he  would  be  in  actual  business. 
(See  Art.  22.) 


COMPLETE  ARITHMETIC. 


94 


(1) 

(2) 

(3) 

(4) 

$1.25 

$19.50 

$75.50 

$1912.88 

8.14 

20.00 

184.30 

806.40 

2.75 

12.45 

111.10 

1000.00 

.65 

14.52 

43.95 

1250.86 

.75 

25.48 

263.55 

943.82 

8.37 

40.50 

100.00 

607.55 

12.50 

8.60 

90.00 

400.33 

4.65 

9.35 

7.15 

148.67 

.83 

.65 

13.48 

249.50 

7.16 

.73 

2.75 

2040.00 

10.28 

.84 

52.30 

4508.70 

1.20 

12.10 

900.25 

3406.30 

.95 

.86 

625.80 

1280.75 

.48 

.93 

314.87 

1300.00 

13.47 

2.95 

64.50 

877.77 

23.00 

14.63 

49.87 

620.14 

3.08 

9.82 

302.58 

8.60 

6.15 

12.60 

10.10 

7.45 

24.92 

19.30 

100.98 

13.33 

.83 

22.33 

78.60 

286.45 

.92 

9.81 

44.50 

1300.80 

.45 

8.76 

77.88 

1440.00 

14.86 

12.57 

320.65 

986.70 

5.80 

18.19 

19.10 

87.80 

7.26 

7.63 

8.50 

137.40 

12.00 

14.60 

436.75 

1500.00 

5.00 

4.85 

135.20 

885.73 

4.37 

9.63 

44.88 

236.40 

6.45 

12.83 

65.90 

13483.86 

17.83 

18.10 

6.01 

11456.20 

2.65 

7.63 

7.83 

88.00 

1.50 

2.20 

4.22 

24.30 

.85 

.35 

3.25 

16.50 

12.20 

.75 

.85 

9.85 

4.65 

8.50 

.62 

100.00 

8.15 

4.65 

1.25 

40.60 

Suggestion. — The  partial  footings  obtained  by  each  summary, 
should  be  written  upon  a  separate  piece  of  paper.  This  will  permit 
the  re-adding  of  any  column  or  set  of  columns,  as  the  case  may  be, 
without  the  trouble  of  re-adding  the  preceding  columns,  and  it  will 
also  avoid  the  defacing  of  the  page  by  erasures  and  corrections. 


ALIQUOTS. 


95 


ALIQUOT  PARTS. 

136.  When  the  price  of  an  article  is  an  aliquot  part  of  a 
dollar,  the  cost  of  any  number  of  such  articles  may  be 
found  more  readily  than  by  multiplying. 

137.  The  aliquot  parts  of  a  dollar  commonly  used  in  busi¬ 
ness,  are : 


50  cts.  =  \ 

of  $1.00 

12|  cts. 

—  i 

—  j 

of  $1.00 

25 

«  —  i 
—  4 

of 

1.00 

6\  “ 

_  i 

—  TS 

of 

1.00 

20 

U  -  1 

—  3 

of 

1.00 

331  « 

—  1 
—  ? 

of 

1.00 

10 

u  —  l 
—  TIT 

of 

1.00 

16f  “ 

—  1 
—  -S' 

of 

1.00 

The  following  aliquot  parts  of  aliquot  parts  of  a  dollar 
are  frequently  used: 

25  cts.  =  |  of  50  cts.  16f  cts.  =  \  of  33^  cts. 

12\  “  =  \  of  50  “  12|  “  =  \  of  25  “ 

6£  “  =  \  of  50  “  6\  “  =  l  of  25  “ 

MENTAL  PROBLEMS. 

1.  What  will  56  pounds  of  grapes  cost,  at  12^-  cts.  a  pound  ? 

Solution. — At  $1  a  pound,  56  pounds  will  cost  $56,  and  at  12| 
cts.,  which  is  |  of  $1,  56  pounds  will  cost  £  of  $56,  which  is  $7. 

2.  What  will  120  spellers  cost,  at  25  cts.  apiece?  At 
33^  cts.  ? 

3.  What  is  the  cost  of  96  dozens  of  eggs,  at  16f  cts.  a 
dozen  ?  At  20  cts.  ?  At  25  cts.  ? 

4.  What  will  240  pounds  of  sugar  cost,  at  12^  cts.  a 
pound?  At  16f  cts. ?  At  20  cts. ? 

5.  At  16|  cents  a  dozen,  how  many  dozens  of  eggs  can 
be  bought  for  $15? 

Solution. — At  16f  cents  a  dozen,  $1  will  buy  6  dozens  of  eggs, 
and  $15  will  buy  15  times  6  dozens,  or  90  dozens. 

6.  At  12^-  cts.  a  pound,  how  many  pounds  of  lard  can 
be  bought  for  $12?  For  $25? 

7.  How  many  pounds  of  butter,  at  33^  cts.  a  pound,  can 
be  bought  for  $15?  For  $33? 


96 


COMPLETE  ARITHMETIC. 


8.  At  6J  cts.  a  quart,  how  many  quarts  of  currants  can 
be  bought  with  30  quarts  of  cherries,  at  10  cts.  a  quart? 


WRITTEN  PROBLEMS. 


9.  What  will  348  yards  of  carpeting  cost,  at  $1.62|-  cts. 
a  yard? 


Process. 


$1.62|  =  $1  +  50  cts.  +  12£  cts. 


$348  =  cost  at  $1  a  yard. 

|  174  =  “  “  50  cts.  a  yard. 

4  43.50  =  “  “  12J“ 

$565.50  =  “  “  $1.62|  “ 

10.  What  will  1600  bushels  of  oats  cost,  at  37^-  cts.  a 
bushel?  At  45  cts.  a  bushel?  At  62^  cts.? 

11.  What  will  2464  bushels  of  wheat  cost,  at  $1.25  a 
bushel?  At  $1.37i?  At  $1.62£? 

12.  What  will  1250  yards  of  carpeting  cost,  at  $1.37^ 
a  yard?  At  $1.50?  At  $1.87-4? 

13.  What  will  640  bottles  of  ink  cost,  at  87^-  cents  a 
bottle?  At  62 J  cts.?  At  75  cts.? 

14.  At  25  cts.  a  dozen,  how  many  dozens  of  eggs  can  be 
bought  for  $42  ?  For  $105?  For  $60.50? 

15.  At  33 J  cts.  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $750?  For  $120? 

16.  What  will  5  lb.  10  oz.  of  butter  cost,  at  35  cts.  a 
pound  ? 

Process. 


$  .35  =  cost  of  1  lb. 

$L75  ==  “  “  5  “ 

.175  =  “  “  8  oz.  lb.) 

.044  =  “  “  2  “  (4  lb.) 

$1,969  =  “  “  5  lb.  10  oz. 


17.  What  will  9  lb.  13  oz.  of  cheese  cost,  at  15  cts.  a 
pound  ?  At  18  cts.  ?  At  20  cts.  ? 

18.  What  will  16  gal.  3  qt.  of  sirup  cost,  at  $1.75  a 
gallon?  At  $1,621?  At  $1.90? 


BILLS. 


97 


19.  What  will  7  bu.  3  pk.  4  qt.  of  cherries  cost,  at  $4.25 
a  bushel  ?  At  $3.50  ?  At  $4.50  ? 

20.  What  will  2  pk.  7  qt.  of  chestnuts  cost,  at  $3.50  a 
bushel?  At  $2.75?  At  $2.62^?  At  $3.12J? 


DEFINITION  AND  RULES. 

138.  An  Aliquot  Part  of  a  number  is  any  integer  or 
mixed  number  which  will  exactly  divide  it. 

139.  Rules. — 1.  To  find  the  cost  of  a  number  of  articles 
when  the  price  is  an  aliquot  part  of  a  dollar,  Find  the  cost 
at  $1,  and  take  such  part  of  the  result  as  the  price  is  of  $1. 

2.  To  find  the  number  of  articles  which  can  be  purchased 
for  a  given  sum  of  money  when  the  price  is  an  aliquot  part 
of  a  dollar,  Find  the  number  of  articles  that  can  be  purchased 
for  $1,  and  multiply  the  result  by  the  given  sum  of  money. 


BILLS. 


140.  Each  of  the  following  bills  should  be  neatly  made 
out  on  paper,  in  proper  form,  and  receipted. 


Thomas  Knight, 
1869 


(1) 

Cincinnati,  O.,  Jan.  1,  1870. 
Bought  of  Baker,  Smith  &  Co. 


a 


u 


u 


u 


18,  48  lb.  Castile  Soap,  @  16|c. 

.  $8.00 

“  25  “  Starch, 

@>  61 

1.56 

30,  65  “  Sugar, 

@i  15 

.  .  .  9.75 

“  33  gal.  Vinegar, 

@r  20 

.  .  6.60 

12,  16  lb.  Rio  Coffee, 

@  23 

3.68 

5  “  Star  Candles,  @  20 

1.00 

“  56  “  Butter, 

@  33j 

.  18.67 

15,  10  “  Cheese, 

@  15 

1.50 

$50.76 

Received  Payment , 


Baker,  Smith  &  Co. 
per  Coons. 


C.Ar.— 9. 


98 


COMPLETE  ARITHMETIC. 


James  Cooper  &  Bro., 


(2) 

St.  Louis,  March  3,  1870. 


To  Charles  Camp  &  Co.,  Dr. 


To  37  bis.  Flour,  Ex.,  @  $4.50  .  .  .  .  $ 

"  23  “  “  Fy.,  @  5.25 

“  25  “  Green  Apples,  2.12|  .... 

“  14  bxs.  Lemons,  @  7.50 

“  5  “  Raisins,  @  4.75  .... 

$ 

Received  Payment , 

What  is  the  amount  due  ? 


(3) 

Cleveland,  O.,  Nov.  24,  1869. 

Dr.  William  Jones, 

To  Charles  C.  Wilhelm,  Dr. 

To  24  Days’  Work,  @  $2.75  .  .  .  .  $ 

“  21  lb.  Nails,  @  6| 

“  540  ft.  Pine  Lumber,  @  4.50  per  100  . 

“  4  M.  Shingles,  @  8.33^-  .  .  .  _ 


By  Cash,  Oct.  16,  .  .  .  .  .  .  $25 

<'  “  “  23, . 44 

il  “  Medical  Services  to  date  .  .  .15 


Received  Payment ,  per  due-bill , 

Charles  C.  Wilhelm. 

What  is  the  amount  of  the  due-bill  ? 

4.  Henry  Smith  bought  of  John  Clarke,  of  Louisville, 
Ky.(  as  follows:  Mch.  10,  1870,  7  pair  calf  boots  @  $5.75; 
6  pair  ladies’  gaiters  @  $3.25;  10  pair  children’s  shoes  @ 
$1.75;  Apr.  1st,  12  pair  coarse  boots  @  $3.12^;  6  pair  calf 


BILLS. 


99 


shoes  @  83.30;  Apr.  12,  7  pair  ladies’  slippers  @  81.33|;  3 
pair  calf  boots  @  85.62^.  Make  out  and  receipt  the  above 
bill  as  clerk  of  John  Clarke. 

5.  Robert  Sterns  &  Co.  bought  of  Dudley  &  Bro.,  Detroit, 
Mich.,  Dec.  20,  1869,  as  follows:  5  doz.  ink-stands  @ 
$2.12^;  9  boxes  steel  pens  @  8.87-J-;  8  reams  note  paper  @ 
83.50;  5  dozen  spellers  @  82.33-J-;  and  2  dozen  copy  books 
@  81.80.  They  sold  Dudley  &  Bro.  3  sets  outline  maps  @ 
88.25,  and  paid  them  815  in  money.  Make  out  the  above 
bill  and  receipt  by  due-bill. 

6.  Mrs.  C.  B.  Jones  bought  of  Cole,  Steele  &  Co.,  of 
Indianapolis,  as  follows:  Nov.  12,  1869,  23  yds.  calico  @ 
16|c. ;  45  yds.  sheeting  @  20c.;  Dec.  7th,  12  yds.  silk  @ 
81.62-J-;  8  handkerchiefs  @  45c.;  2  pair  kid  gloves  @  81.87J. 
Make  out  and  receipt  the  above  bill. 

DEFINITIONS. 

141.  An  Account  is  a  record  of  business  transactions 
between  two  parties,  with  specifications  of  debts  and  credits. 

The  party  owing  the  debts  specified,  is  called  the  Debtor , 
and  the  party  to  whom  they  are  due,  is  called  the  Creditor. 

142.  A  Bill  is  a  written  statement  of  an  account.  It  is 
drawn  by  the  creditor  against  the  debtor,  and  gives  the  time 
and  place  of  the  transaction,  and  the  names  of  the  parties. 

When  the  debtor  has  made  payments  on  the  account,  or 
has  charges  against  the  creditor,  such  payments  or  charges 
are  called  Credits.  They  are  entered  as  in  Bill  3. 

143.  A  bill  is  receipted  by  writing  the  words  “ Received 
Payment”  at  the  bottom,  and  affixing  the  creditor’s  name. 
This  may  be  done  by  the  creditor,  or  by  a  clerk,  agent,  or 
any  other  authorized  person. 

If  the  debtor  is  not  able  to  pay  a  bill  when  presented,  it 
may  be  accepted  by  writing  the  word  “ Accepted  ”  across  its 
face,  with  date  and  signature.  When  a  bill  is  paid  by  a 


100 


COMPLETE  ARITHMETIC. 


promissory  note  or  due-bill,  the  fact  may  be  added  to  the 
words  “Received  Payment”  as  in  Bill  3. 

144.  A  Hill  of  Goods  is  a  written  statement  of  goods 
sold,  with  the  amount  and  price  of  each  article,  and  the 
entire  cost.  It  is  also  called  an  Invoice. 

When  sales  are  made  at  different  times,  the  date  is 
written  at  the  left,  as  in  Bill  1. 


SECTION  XI. 


MENSURATION 


I.  SURFACES. — Definitions. 


145.  A  Line  is  length.  _  _ 

146.  A  Straight  Line  is  a  line  having  the  same 

direction  throughout  its  whole  exent.  _ 

Note. — The  word  line  is  commonly  used  to  denote  a  straight  line. 

147.  An  Angle  is  the  divergence  of  two  lines  which  have 
a  common  point.  The  common  point  is  called  the  vertex. 


Thus  the  divergence  of  the  lines  B  A 
and  B  C  is  the  angle  ABC,  and  the  point 
B  is  its  vertex. 


148.  When  a  line  so  meets  another  line  as  to  make  the 
two  adjacent  angles  equal,  each  angle  is  a  Bight  Angle,  and 
the  first  line  is  perpendicular  to  the  second. 


A 


Thus  the  two  equal  adjacent  angles 
ABC  and  A  B  D  are  right  angles,  and 
the  line  A  B  is  perpendicular  to  the  line 


CD. 


MENSURATION 


101 


149.  An  Obtuse  Angle  is  greater  than  a  right  angle, 
and  an  Acute  Angle  is  less  than  a  right  angle. 

Thus  the  angle  A  BD  is  an  obtuse 
angle,  and  the  angle  ABC  is  an  acute 
angle.  The  line  A  B  is  an  oblique  line. 

150.  A  Surface  is  that  which  has  length  and  width, 
but  not  depth  or  thickness. 

151.  A  j Plane  Surface  is  a  surface  such  that  all 
possible  straight  lines  connecting  each  two  points  of  it,  lie 
wholly  within  the  surface.  It  is  also  called  a  Plane. 

Note —To  determine  whether  the  surface  of  a  table  is  a  plane, 
take  a  ruler  with  a  straight  edge  and  apply  it  to  the  surface  in  many 
different  directions.  If  the  edge  rests  uniformly  upon  the  surface,  it 
is  a  plane. 

152.  A  Rectangle  is  a  plane  ^  -  ^ 

figure  bounded  by  four  straight  lines  -  BH  IBS 

and  having  four  right  angles.  ■  H 

153.  A  Square  is  a  rectangle  with  its  four  sides  equal. 

A  Square  Inch  is  a  square  i  inch, 

each  side  of  which  is  an  inch  in 
length. 

The  figure  represents  a  square  inch  of  c 
real  size.  - 

A  square  foot ,  square  yard,  square  rod,  etc., 
are  squares  whose  sides  are  respectively  1 
foot,  1  yard,  1  rod,  etc.,  in  length.  linch. 


154.  A  Triangle  is  a  plane  fig¬ 
ure  bounded  by  three  straight  lines 
and  having  three  angles. 


102 


COMPLETE  ARITHMETIC. 


Sh' 

o5 

'a 

•3 

p 

Cj 


u 

QJ 


Pi 


.Base. 


155.  A  Right-angled  Tri¬ 
angle  is  a  triangle  having  a  right 
angle.  One  of  the  sides  including 
the  right  angle  is  called  the  Base, 
and  the  other  the  Perpendicular  or 
Altitude. 


156.  A  Circle  is  a  portion  of  a  plane  bounded  by  a 

curved  line,  all  points  of  which  are 
equally  distant  from  a  point  within, 
called  the  center. 

The  curved  line  which  bounds  a 
circle  is  its  Circumference. 


One-half  of  a  circumference  is  a  Semi- 
circumference;  one-fourth  is  a  Quadrant ; 
and  any  portion  is  an  Arc. 

157.  The  Diameter  of  a  circle  is  a  straight  line 
passing  through  the  center  and  terminating  on  both  sides 

in  the  circumference.  One-half  of  a  diameter  is  a  Radius. 

\ 

All  the  diameters  of  a  circle  are  equal,  and  all  the  radii  are 
equal. 

The  circumference  of  a  circle  is  3.1416  (nearly  3})  times  the 
diameter. 


158.  The  Area  of  a  plane  figure  is  its  extent  of  surface, 
or  superficial  contents.  It  is  expressed  by  some  unit  of 
measure,  as  a  square  inch,  a  square  foot,  etc. 


159.  The  area  of  a  right-angled 
triangle  is  one-half  the  area  of  a 
rectangle  with  the  same  base  and 
altitude.  The  triangle  A  B  C  is  one- 
half  of  the  rectangle  A  B  C  D. 


160.  The  area  of  a  circle  equals  the  product  of  the  cir¬ 
cumference  by  the  one-half  of  the  radius. 

Note. — This  may  be  illustrated  by  dividing  a  circle  by  diameters 
into  eighths,  and  considering  each  a  triangle. 


MENSURATION, 


103 


MENTAL  PROBLEMS. 

1.  How  many  square  inches  in  a  piece  of  paper  4  inches 
long  and  1  inch  wide?  4  inches  long  and  2  inches  wide? 

2.  How  many  square  feet  in  a  piece  of  zinc  4  feet  long 
and  1  foot  wide  ?  3  feet  wide  ?  4  feet  wide  ? 

3.  How  many  square  inches  in  a  pane  of  glass  12  inches 
square?  Then  how  many  square  inches  in  a  square  foot? 

4.  How  many  square  feet  in  a  piece  of  oil-cloth  7  feet 
long  and  3  feet  wide?  8  ft.  long  and  6  ft.  wide? 

5.  How  many  square  feet  in  a  square  yard? 

6.  How  many  square  feet  in  the  floor  of  a  room  20  by  15 
ft.  ?  30  by  24  ft.  ?  50  by  30  ft.  ? 

Note. — The  dimensions  of  a  plane  figure  are  usually  expressed  by 
writing  the  word  “  by,”  or  the  sign  “  X,”  between  the  figures  denoting 
the  length  and  width. 

7.  How  many  square  yards  in  a  pavement  40  by  5  yd.  ? 
50  X  4  yd.  ?  80  X  5  yd.  ? 

8.  How  many  square  miles  in  a  township  5  miles  square? 
6  miles  square? 

9.  How  many  square  inches  in  a  right-angled  triangle, 
whose  base  is  8  inches  and  whose  altitude  is  6  inches? 

10.  The  diameter  of  a  circle  is  10  feet:  what  is  its  cir¬ 
cumference  ? 


WRITTEN  PROBLEMS. 

11.  How  many  square  feet  in  a  floor  37 J  by  23  ft.? 

Process:  37|  sq.  ft.  X  23  =  862|  sq.  ft. 

12.  How  many  square  yards  in  a  walk  124.5  by  3.25  yd.? 

13.  How  many  square  feet  in  the  walls  of  a  room  24  by 
18J  ft.  and  101  ft.  high?  What  is  the  area  of  the  ceiling? 

14.  How  many  square  chains  in  a  farm  134  chains  long 
and  52.5  chains  wide? 

15.  How  many  square  feet  in  a  city  lot  62^  ft.  by  208  ft.  ? 

16.  A  garden  containing  3267  square  yards  is  49 J  yards 
wide:  how  long  is  it? 


104 


COMPLETE  ARITHMETIC. 


17.  A  street  containing  800  square  rods  is  33^  rods  long: 
how  wide  is  it? 

18.  How  many  yards  of  carpeting,  J  of  a  yard  wide,  will 
cover  a  room  15  by  8^  yd.  ? 

19.  How  many  square  yards  in  a  triangular  garden  whose 
base  is  54.5  yards,  and  altitude  33.2  yards? 

20.  A  triangle  contains  270  sq.  in.,  and  the  base  is 
36  in. :  what  is  its  altitude  ? 

21.  The  diameter  of  a  circle  is  12  inches:  how  many 
square  inches  in  its  area  ? 

22.  How  many  square  feet  in  a  circle  20  ft.  in  diam¬ 
eter? 

161.  Rules. — 1.  To  find  the  area  of  a  rectangle,  Multiply 
the  length  by  the  width. 

2.  To  find  either  side  of  a  rectangle,  Divide  the  area  by  the 
other  side. 

3.  To  find  the  area  of  a  triangle,  Multiply  the  base  by  one 
half  the  altitude. 

4.  To  find  the  circumference  of  a  circle,  Multiply  the  diam¬ 
eter  by  3.1416. 

5.  To  find  the  area  of  a  circle,  Multiply  the  circumference  by 
one  fourth  of  the  diameter. 


II.  SOLIDS.  — Definitions. 


162.  A  Solid  is  that  which  has  length,  width,  and  depth 
or  thickness.  It  is  also  called  a  Volume  or  Body. 

A  line  has  only  length ;  a  surface  has  length  and  width ;  and  a 
solid  has  length,  width,  and  depth. 


163.  A  Rectangular  Solid  is 
a  body  bounded  by  six  rectangular 
surfaces. 


The  surfaces  bounding  a  solid  are  called  Faces ,  and  the  sides  of 
these  faces  are  called  Edges.  A  rectangular  solid  has  twelve  edges. 
The  face  on  which  a  solid  is  supposed  to  rest  is  called  its  Base. 


MENSURATION. 


105 


164.  A  Cube  is  a  body  bounded 
by  six  equal  squares.  All  its  edges 
are  equal. 

A  Cubic  Inch  is  a  cube  whose 
edges  are  each  one  inch  in  length. 

A  cubic  foot ,  cubic  yard ,  cubic  rod,  etc., 
are  each  cubes  whose  edges  are  respectively 


1  foot,  1  yard,  1  rod,  etc. 


165.  A  Cylinder  is  a  solid  whose  two  bases 
are  equal  and  parallel  circles,  and  whose  lateral 
surface  is  uniformly  curved. 

166.  The  volume  of  a  body  is  called  its  Solid 
Contents ,  or  Capacity.  It  is  expressed  in  some  unit 
of  measure;  as  a  cubic  inch,  a  cubic  foot,  etc. 


MENTAL  PROBLEMS. 


1.  How  many  cubic  inches  in  a 
rectangular  solid,  4  inches  long,  1 
inch  wide,  and  1  inch  thick? 

2.  How,  many  cubic  inches  in  a 
rectangular  solid,  4  inches  long,  3 
inches  wide,  and  1  inch  thick? 

3.  How  many  cubic  inches  in  a 
rectangular  solid,  4  inches  long,  3 
inches  wide,  and  2  inches  thick? 

4.  How  many  cubic  feet  in  a  block 
of  marble  6  ft.  long,  3  ft.  wide,  and 
2  ft.  thick?  10  ft.  long,  5  ft.  wide,  and  4  ft.  thick? 

5.  How  many  cubic  feet  in  a  cubic  yard? 

6.  How  many  cubic  feet  in  a  bin  6  ft.  long,  3  ft.  wide, 
and  3  ft.  deep?  8  ft.  long,  5  ft.  wide,  and  2  ft.  deep? 

7.  How  many  cubic  yards  in  a  room  5  yd.  long,  4  yd. 
wide,  and  3  yd.  high? 


106 


COMPLETE  ARITHMETIC. 


WRITTEN  PROBLEMS. 


8.  How  many  cubic  feet  in  a  block  of  granite  16  ft. 
long,  8  ft.  wide,  and  5  ft.  thick? 


Process. 

16  cu.  ft. 

_8 

128  cu.  ft. 
_5 

640  cu.  ft. 


A  block  16  ft.  long,  1  ft.  thick,  and  1  ft.  wide,  con¬ 
tains  16  cu.  ft. ;  and  a  block  16  ft.  long,  1  ft.  thick, 
and  8  feet  wide,  contains  8  times  16  cu.  ft.,  or  128 
cu.  ft.;  and  a  block  16  ft.  long,  8  feet  wide,  and  5  ft. 
thick,  contains  5  times  128  cu.  ft.  =  640  cu.  ft.  Hence, 
solid  contents  =  16  cu.  ft.  X  8  X  5. 


9.  How  many  cubic  feet  in  a  pile  of  wood  45  ft.  long, 
3^  ft.  wide,  and  7  ft.  high  ? 

10.  How  many  cubic  yards  in  a  cubic  rod? 

11.  How  many  cubic  feet  in  a  cube  each  of  whose  edges 
is  12 \  ft.  in  length? 

12.  A  building,  65  ft.  by  44  ft.,  has  a  foundation  wall 
12  ft.  deep  and  2  ft.  thick:  how  many  cubic  feet  in  the 
foundation  wall? 

13.  A  pile  of  wood,  containing  840  cu.  ft.,  is  30  ft.  long 
and  3 J  ft.  wide  :  how  high  is  the  pile  ? 

14.  If  27  bricks  make  a  cubic  foot,  how  many  bricks  will 
make  a  wall  45  ft.  long,  27  ft.  high,  and  2^  ft.  thick  ? 

15.  How  many  cans,  6  by  4  by  2  in.,  can  be  placed  in  a 
box  30  by  18  by  20  in.  in  the  clear? 

16.  The  base  of  a  cylinder  is  12  inches  in  diameter,  and 
its  altitude  is  25  inches:  how  many  cubic  inches  in  it? 


167.  Rules. — 1.  To  find  the  solid  contents  of  a  rectan¬ 
gular  solid,  Multiply  the  length ,  width ,  and  thickness  together. 

2.  To  find  the  length,  width,  or  thickness  of  a  rectangular 
solid,  Divide  the  solid  contents  by  the  product  of  the  other  two 
dimensions. 

3.  To  find  the  entire  surface  of  a  cylinder,  Multiply  the 
circumference  of  the  base  by  the  altitude ,  and  to  the  product  add 
the  areas  of  the  two  bases. 

4.  To  find  the  solid  contents  of  a  cylinder,  Multiply  the 
area  of  the  base  by  the  altitude. 


REDUCTION. 


107 


SECTION  XII. 

DENOMINATE  NUMBERS. 


REDUCTION. 

Case  I. 

Reduction,  of*  Denominate  Integers  and  IVIixed 

Numbers. 

1.  How  many  mills  in  9  cents?  In  12-J-  cents?  62^- 
cents?  100  cents? 

2.  How  many  cents  in  7  dimes?  25^  dimes?  45.4 
dimes?  56.8  dimes?  75.3  dimes? 

3.  How  many  dollars  in  50  dimes  ?  120  dimes  ?  145 

dimes?  1250  dimes?  1625  dimes? 

4.  How  many  dollars  in  800  cents  ?  2400  cents  ?  1365 

cents  ?  2235  cents  ? 

5.  How  many  farthings  in  9  pence?  72  pence?  90^- 
pence?  24.5  pence? 

Note. — For  tables  see  appendix. 

6.  How  many  pence  in  8J  shillings?  10^  s.  ?  33^  s.  ? 

2.5  s,  ?-  6.5  s.  ? 


108 


COMPLETE  ARITHMETIC. 


7.  How  many  shillings  in  15  £  ?  2.5  £?  16.4  £? 

8.  How  many  pence  in  22  far.  ?  48  far.  ?  105  far.  ? 
201  far.? 

9.  How  many  pounds  in  120  s.  ?  360  s.  ?  720  s.  ? 

10.  How  many  shillings  in  72  d.  ?  144  d.  ?  25.2  d.  ? 

34.8  d.  ?  52.92  d.?  73.44  d.? 

11.  How  many  drams  in  8  oz.  avoir.?  20  oz.  ?  4.5  oz. ? 

12.  How  many  ounces  in  5  lb.  avoir.  ?  10^-  lb.  ?  2.5  lb.  ? 

13.  How  many  pounds  in  64  oz.  avoir.  ?  19.2  oz.  ?  4.8  oz.  ? 

14.  How  many  grains  in  5  pwt.  ?  10^-  pwt.  ?  2.5  pwt.  ? 

15.  How  many  pwt.  in  7  oz.?.  6.5  oz.  ?  12J-  oz. ? 

16.  How  many  ounces  of  gold  in  7  lb.  ?  12 J  lb.  ?  1.5  lb.  ? 
4.5  1b.?  12.5  1b.? 

17.  How  many  pounds  of  gold  in  48  oz.  ?  14.4  oz.  ? 

2.52  oz.?  4.68  oz.?  62.4  oz.? 

18.  How  many  scruples  in  12  3?  8-J  3  ?  14.5  3? 

19.  How  many  drams  in  15  3  ?  12f3?  11.5  3? 

20.  How  many  ounces  in  9  lb  ?  5.5  lb?  10.5  lb? 

21.  How  many  inches  in  8J  ft.?  15|-  ft.?  33L  ft.? 

22.  How  many  yards  in  12  rd.?  1.6  rd.  ?  3.2  rd.  ? 

23.  How  many  rods  in  11  yd.?  33  yd.?  6.6  yd.? 

24.  How  many  miles  in  18  fur.  ?  13.6  fur.?  7.2  fur.? 

25.  How  many  sq.  ft.  in  3£  sq.  yd.  ?  16f  sq.  yd.  ? 

26.  How  many  square  yards  in  12.6  sq.  ft.?  49.5  sq.  ft.? 
1.71  sq.  ft.?  56.7  sq.  ft.? 

27.  How  many  quarts  in  17  pk.  ?  12|  pk.  ?  30^  pk.? 

28.  How  many  gallons  in  35  qt.  ?  14.8  qt.  ?  2.56  qt.  ? 

29.  How  many  weeks  in  365  days?  25.2  days? 

30.  How  many  years  in  192  mo.?  25.2  mo.?  100  mo.? 


WRITTEN  PROBLEMS. 

31.  Reduce  5£  6  s.  3d.  to  pence.  1275  d.  to  pounds. 

12  ) 1275 


Process  : 


5  £  6  s.  3d. 
20 


106  s. 
12 


Process  :  20  )  106  3  d. 

5  £  6  s. 


1275  d.,  Ans. 


5 £  6s.  3d.,  Ans. 


REDUCTION. 


109 


32.  Reduce  38  lb.  11  oz.  7  dr.  to  drams. 

33.  Reduce  12  bu.  5  qt.  to  pints. 

34.  Reduce  13  mi.  5  fur.  3  yd.  to  yards. 

35.  Reduce  11  A.  3  R.  22  P.  to  perches. 

36.  Reduce  503  pt.  to  bushels. 

37.  Reduce  324  gi.  to  gallons. 

38.  Reduce  10280  ft.  to  miles. 

39  Reduce  12460"  to  degrees. 

40.  Reduce  30684  sec.  to  higher  denominations. 

41.  How  many  pence  in  £45?  In  £237§? 

42.  How  many  perches  in  95  A.  ?  320J  A.  ? 

43.  How  many  hundred-weight  in  4085  oz.  avoir.  ? 

44.  How  many  miles  in  12840  ft.  ? 

45.  Reduce  13  mi.  5|-  fur.  to  inches. 

46.  Reduce  113420  inches  to  miles. 

47.  Reduce  3450  cubic  feet  of  wood  to  cords. 

48.  Reduce  5124  quarts  to  bushels. 

49.  Reduce  16  common  years  to  hours. 

50.  How  many  seconds  were  in  the  year  1868  ? 

51.  Reduce  4  common  yr.  45  d.  to  minutes. 

52.  Reduce  3.7  bushels  to  pints. 

53.  Reduce  4.5  rods  to  feet. 

54.  Reduce  3.65  lb.  Troy  to  ounces. 

55.  Reduce  15°  40'  36"  to  seconds. 

56.  Reduce  588487"  to  degrees. 

57.  Reduce  12.3  miles  to  feet. 

58.  Reduce  365J  days  to  weeks. 

59.  Reduce  706.35  perches  to  acres. 

60.  How  many  acres  in  12f  sq.  miles? 

168.  Rules. — I.  To  reduce  a  denominate  number  from 
a  higher  to  a  lower  denomination, 

1.  Multiply  the  number  of  the  ^highest  denomination  by  the 
number  of  iniits  of  the  next  lower  which  equals  a  unit  of  the 
higher,  and  to  the  product  add  the  number  of  the  lower  denomi¬ 
nation,  if  any. 

2.  Proceed  in  like  manner  with  this  and  each  successive 


110 


COMPLETE  ARITHMETIC. 


result  thus  obtained ,  until  the  number  is  reduced  to  the  required 
denomination. 

Note. — The  successive  denominations  of  the  compound  number 
should  be  written  in  their  proper  orders,  and  the  vacant  denomina¬ 
tions,  if  any,  filled  with  ciphers. 

II.  To  reduce  a  denominate  number  from  a  lower  to  a 
higher  denomination, 

1.  Divide  the  given  denominate  number  by  the  number  of 
units  of  its  own  denomination  which  equals  one  unit  of  the  next 
higher,  and  place  the  remainder,  if  any,  at  the  right. 

2.  Proceed  in  like  manner  with  this  and  each  successive 
quotient  thus  obtained,  until  the  number  is  reduced  to  the  re¬ 
quired  denomination. 

3.  The  last  quotient,  with  the  several  remainders  annexed  in 
proper  order,  will  be  the  answer  required. 

Note. — The  above  rules  also  apply  to  the  reduction  of  denominate 
fractions,  both  common  and  decimal.  (Art.  169.) 

Case  II. 

Reduction  of  Denominate  Fractions. 

1.  What  part  of  a  peck  is  T*g-  of  a  bushel?  bu.  ? 

Solution. —  Th  bu.  =  Tag  of  4  pk.  =  x4g  pk.  or  \  pk.,  and  bu.  = 
3  times  \  pk.  =  f  pk.  Hence,  T3g  bu.  =  §  pk. 

2.  What  part  of  a  quart  is  of  a  peck  ?  ^  pk.  ? 

3.  What  part  of  a  day  is  T2-  of  a  week  ?  T87  w.  ? 

4.  What  part  of  an  hour  is  ^  of  a  day?  A  d*  ? 

5.  What  part  of  an  inch  is  of  a  foot  ?  ^  ft.  ? 

6.  What  decimal  part  of  an  inch  is  .03  of  a  foot? 

Solution. —  .03  ft.  =  .03  of  12  in.,  or  12  times  .03  in.  =  .36  in. 

7.  What  decimal  of  an  hour  is  .05  of  a  day?  .025  d.  ? 

8.  What  decimal  of  a  day  is  .12  of  a  week?  .012  w.  ? 

9.  What  decimal  of  a  quart  is  .125  of  a  peck?  .35  pk.  ? 

10.  What  part  of  an  inch  is  ^  of  a  foot?  .08  ft.  ? 

11.  What  part  of  a  pint  is  x30  of  a  gallon?  .06  gal.  ? 


REDUCTION. 


Ill 


12.  What  part  of  a  foot  is  §  of  an  inch? 

Solution. —  |  in.  =  f  of  TJZ  ft.  =  ^  ft. 

13.  What  part  of  a  week  is  of  a  day?  ^  d.? 

14.  What  part  of  an  hour  is  -fa  of  a  minute?  ^T°-  min.  ? 

15.  What  part  of  a  gallon  is  f  of  a  pint?  f  pt.  ? 

16.  What  part  of  a  pound  avoir,  is  f  of  an  ounce? 

17.  What  decimal  of  a  foot  is  .48  of  an  inch? 

Solution. —  .48  in.  —  .48  of  ft.  =  fa  of  .48  ft.  =  .04  ft. 

18.  What  decimal  of  a  bushel  is  .12  of  a  peck?  3.6  pk.  ? 

19.  What  decimal  of  a  week  is  .49  of  a  day?  6.3  d.  ? 

20.  What  decimal  of  a  pound  Troy  is  .144  of  an  ounce? 
2.52  oz. ?  38.4  oz.  ?  .72  oz.  ?  9.6  oz.  ? 

21.  What  decimal  of  a  ream  is  .8  of  a  quire?  2.8  quires? 

22.  What  part  of  a  dime  is  f  of  a  cent?  .625  ct. ? 

23.  What  part  of  a  shilling  is  f  of  a  penny?  .6  d.  ? 
.18  d.?  2.4  d.?  1.44  d.? 

24.  What  part  of  a  gallon  is  of  a  pint?  .64  pt. ? 


WRITTEN  PROBLEMS. 


25.  Reduce  rs of  a  day  to  the  fraction  of  a  minute. 


Process  : 


7  d  =  7_X  24  h  =  7  X  24  X  60 
18000  '  18000  *  18000 


min. 


14  . 

—  min. 

25 


26.  Reduce  -5-^75-  of  a  pound  avoirdupois  to  the  fraction 
of  a  dram. 

27.  Reduce  \\  of  a  yard  to  inches. 

28.  Reduce  of  a  pound  Troy  to  pennyweights. 

29.  Reduce  .005  of  a  pound  to  the  decimal  of  a  penny. 

30.  Reduce  .0065  of  a  week  to  the  decimal  of  an  hour. 

31.  Reduce  9.6  pwt.  to  the  decimal  of  a  pound  Troy. 

32.  Reduce  3.96  inches  to  the  decimal  of  a  rod. 

33.  Reduce  30.8  rods  to  the  decimal  of  a  mile. 

34.  Reduce  .096  of  a  bushel  to  the  decimal  of  a  pint. 

35.  Reduce  of  a  rod  to  the  fraction  of  a  league. 


112 


COMPLETE  ARITHMETIC. 


36.  Reduce  |-f  of  a  degree  to  the  fraction  of  a  circum¬ 
ference. 

37.  Reduce  -J-J-  of  a  day  to  minutes. 

38.  Reduce  of  a  week  to  hours. 

39.  Reduce  f  of  a  minute  to  the  fraction  of  a  day. 

40.  Reduce  11.2  perches  to  the  decimal  of  an  acre. 

41.  Reduce  13.62  cords  to  cord  feet. 

42.  Reduce  .037  lb.  avoirdupois  to  drams. 

43.  Reduce  56f  lb.  Troy  to  grains. 

44.  Reduce  of  a  gallon  to  the  fraction  of  a  pint. 

45.  Reduce  2.43  miles  to  feet. 

46.  Reduce  777.6  pence  to  pounds. 

47.  Reduce  1.408  ft.  to  the  decimal  of  a  mile. 

48.  Reduce  ^  of  an  hour  to  the  fraction  of  a  day. 

49.  Reduce  .012  of  a  mile  to  yards. 

50.  Reduce  f  of  a  yard  to  the  decimal  of  a  mile. 

169.  Rule. — To  reduce  denominate  fractions  from  a 
higher  to  a  lower  denomination,  or  from  a  lower  to  a 
higher,  Proceed  as  in  the  reduction  of  denominate  integers . 

Note. — Denominate  fractions  are  reduced  to  a  lower  denomination 
by  multiplying,  and  to  a  higher  denomination  by  dividing,  the  same 
as  denominate  integers ;  but  in  reduction  descending  there  arc  no 
units  of  a  lower  order  to  add,  and  in  reduction  ascending  there  are 
no  remainders. 


Case  III. 

Reduction  of  Denominate  Fractions  to  Lower 

Integers. 

1.  How  many  months  in  |  of  a  year?  J  of  a  year? 
|  of  a  year? 

2.  How  many  hours  in  f  of  a  day?  J  of  a  day?  i-J 
of  a  day? 

3.  How  many  minutes  in  IT  of  an  hour?  of  an 
hour?  yu  of  an  hour? 

4.  How  many  yards  in  of  a  rod?  -§  of  a  rod? 
of  a  rod  ? 

5.  How  many  quarts  in  .75  of  a  peck?  1.25  pk.  ? 


REDUCTION. 


113 


6.  How  many  months  in  .25  of  a  year?  .33^  yr.  ? 

7.  How  many  days  in  .35  of  a  week?  4.5  w. ?  7.3  w. ? 

8.  How  many  pecks  and  quarts  in  .85  of  a  bushel? 

Solution. —  .85  bu.  =  .85  of  4  pk.  =  3.4  pk.,  and  .4  pk.  =  .4  of 
8  qt.  =  3.2  qt.  Hence,  .85  bu.  =  3  pk.  3.2  qt. 

9.  How  many  feet  and  inches  in  .75  of  a  yard? 

10.  How  many  quarts  and  pints  in  f  of  a  gallon? 

11.  How  many  days  and  hours  in  -J  of  a  week? 

12.  How  many  pecks  and  quarts  in  .55  of  a  bushel? 


WHITTEN  PROBLEMS. 


13.  Reduce  ^  of  a  day  and  .415  of  an  hour  each  to 
integers  of  lower  denominations. 


Process.  Process. 

7  V  24  A15  h. 

&  da.  =  TV  of  24  h.  =  h.  =  10|  h.  _60 

16  24.900  min. 

\  h.  =  i  of  60  min.  =  —  min.  =  30  min.  _ §9 

2  54.000  sec. 


■fo  da.  =  10  h.  30  min. 


.415  h.  =  24  min.  54  sec. 


Reduce  to  integers  of  lower  denominations 


14.  |  of  a  mile. 

15.  of  a  week. 

16.  of  a  lb.  Troy. 

17.  of  a  rod. 

18.  of  an  acre. 

19.  {  of  a  cord. 


20.  .85  of  a  lb.  avoir. 

21.  .325  of  a  ton. 

22.  .08^  of  a  yard. 

23.  .9375  of  a  gallon. 

24.  .5625  of  a  cwt. 

25.  .0135  of  a  cord. 


170.  Rule. — To  reduce  a  denominate  fraction  to  inte 
gers  of  lower  denominations, 

1.  Multiply  the  fraction  by  the  number  of  units  of  the  next 
lower  denomination,  which  equals  a  unit  of  its  denomination. 

2.  Proceed  in  like  manner  with  the  fractional  part  of  the 
product  and  of  each  succeeding  product,  until  the  lowest 

denomination  is  reached. 

C.Ar.— 10. 


114 


COMPLETE  ARITHMETIC. 


3.  The  integral  parts  of  the  several  products,  written  in 
proper  order,  will  he  the  lower  integers  sought . 

Note. — When  the  last  product  contains  a  fraction,  it  should  be 
united  with  the  integer  of  the  lowest  denomination,  forming  a  mixed 
number. 

Case  IV. 

He  cl  notion  of  Integers  of  Lower  Denominations 
to  Fractions  of  Higher  Denominations. 

1.  What  part  of  a  dollar  is  25  cents?  50  cts. ? 

2.  What  part  of  a  foot  is  8  inches  ?  10  in.  ? 

3.  What  part  of  a  day  is  9  hours?  15  h. ? 

4.  What  part  of  a  yard  is  2  ft.  6  in.  ? 

Solution. —  1  yd.  =  36  in.,  and  2  ft.  6  in.  =  30  in. ;  1  in.  =  of 
a  yd.,  and  30  in.  =  §£  yd.  =  f  yd.  Hence,  2  ft.  6  in.  =  •§  yd. 

5.  What  part  of  a  gallon  is  3  qt.  1  pt.  ? 

6.  What  part  of  a  bushel  is  2  pk.  5  qt.  ? 

7.  What  part  of  a  rod  is  3  yd.  2  ft.  ? 

8.  What  part  of  a  barrel  (31  gal.)  is  15  gal.  2  qt.  ? 

9.  What  part  of  3  pecks  is  2  pk.  4  qt.  ? 

10.  What  part  of  5  yards  is  2  yd.  2  ft.  ? 

Suggestion. — Each  of  the  above  answers  should  be  expressed 
both  as  a  common  fraction  and  as  a  decimal. 

WHITTEN  PROBLEMS. 

11.  Reduce  15  w.  5  da.  to  the  fraction  of  a  common  year. 

Process. 

15  w.  5  da.  =  110  da. 

yr-  =  ft  yr->  ^s. 

12.  Reduce  1  yd.  2  ft.  6  in.  to  the  fraction  of  a  rod. 

13.  Reduce  1  pk.  2  qt.  1-J-  pt.  to  the  fraction  of  a  bushel. 

14.  Reduce  9  oz.  2^  dr.  to  the  fraction  of  a  pound. 

15.  Reduce  9  h.  36  min.  to  the  decimal  of  a  year. 

16.  Reduce  2  pk.  3  qt.  1.2  pt.  to  the  decimal  of  a  bushel. 

17.  Reduce  13  s.  4  d.  to  the  decimal  of  a  pound  Sterling. 


REDUCTION. 


115 


18.  Reduce  1  R.  14  P.  to  the  decimal  of  an  acre. 

19.  Reduce  8  oz.  8  pwt.  to  the  decimal  of  a  pound  Troy. 

20.  Reduce  1  fur.  18  rd.  1  yd.  to  the  decimal  of  a  mile. 

21.  What  part  of  1  bu.  3  pk.  is  5  pk.  6  qt.  ? 

22.  What  part  of  3  w.  4  da.  is  3  da.  8  h.  ? 

23.  What  part  of  12  A.  2  R.  is  1  A.  2  R.  10  P.  ? 

24.  What  part  of  3  barrels  of  flour  is  110  lb.  4  oz.  ? 

171.  Rule. — To  reduce  a  denominate  number,  simple 
or  compound,  to  the  fraction  of  a  higher  denomination. 
Reduce  the  number  which  is  a  part  and  the  number  which  is  a 
whole  to  the  same  denomination ,  and  write  the  former  result  as 
a  numerator  and  the  latter  as  a  denominator  of  a  fraction. 

Notes. — 1.  The  answer  may  be  expressed  decimally  by  changing 
the  common  fraction  to  a  decimal. 

2.  When  the  whole  is  a  unit  and  the  part  a  compound  number, 
the  process  may  be  somewhat  shortened  by  reducing  the  number  of  the 
lowest  denomination  to  a  fraction  of  the  next  higher,  prefixing  the  higher 
number,  if  any,  and  then  reducing  this  result  to  a  fraction  of  the  next 
higher  denomination,  and  so  on,  until  the  required  fraction  is  reached. 
Thus,  in  the  16th  problem  above,  the  1.2  pt.  =  .6  qt. ;  and  3.6  qt.  = 
.45  pk.;  and  2.45  pk.  =  .6125  bu. 

DEFINITIONS. 

172.  A  Denominate  Number  is  a  number  com¬ 
posed  of  concrete  units  of  one  or  several  denominations. 
It  may  be  an  integer,  a  mixed  number,  or  a  fraction. 

173.  Denominate  numbers  are  either  Simple  or  Compound. 

A  Simple  Denominate  Number  is  composed  of 

units  of  the  same  denomination ;  as,  7  quarts. 

A  Compound  Denominate  Number  is  com¬ 
posed  of  units  of  several  denominations;  as,  5  bu.  3  pk.  7  qt. 
It  is  also  called  a  Compound  Number. 

Note. — Every  compound  number  is  necessarily  denominate. 

174.  Denominate  numbers  express  Currency ,  Measure ,  and 
Weight. 

Currency  is  the  circulating  medium  used  in  trade  and 
commerce  as  a  representative  of  value. 


116 


COMPLETE  ARITHMETIC. 


JKeasuve  is  the  representation  of  extent,  capacity,  or 
amount. 

TVeight  is  a  measure  of  the  force  called  gravity,  by 
which  bodies  are  drawn  toward  the  earth. 

175.  The  following  diagram  represents  the  three  general 
classes  of  denominate  numbers,  their  subdivisions,  and  the 
tables  included  under  each : 


I. 


Currency, 


{ 


1. 

2. 


Coin,  |  f  1. 

Paper  Money.  /  1  2. 


United  States  Money, 
English  Money. 


1.  Lines  |  L  LonS  Measure, 
and  arcs,  1  %  Circular  Measure 


II.  Measure, 


1.  Of  extension, 


2.  Surfaces:  Square  Measure. 


_  3.  Capacity, 


1.  Cubic  Measure, 

2.  Wood  Measure, 

3.  Dry  Measure, 

4.  Liquid  Measure. 


2.  Of  duration :  Time  Measure. 


{1.  Avoirdupois  Weight, 
2.  Troy  Weight, 

3.  Apothecaries  Weight. 


Note. — For  tables  see  appendix. 

176.  The  Reduction  of  a  denominate  number  is  the 
process  of  changing  it  from  one  denomination  to  another 
without  altering  its  value. 

177.  Reduction  is  of  two  kinds,  Reduction  Descending  and 
Reduction  Ascending. 

Reduction  Descending  is  the  process  of  changing 
a  denominate  number  from  a  higher  to  a  lower  denomi¬ 
nation. 

Reduction  Ascending  is  the  process  of  changing 
a  denominate  number  from  a  lower  to  a  higher  denomi¬ 
nation. 


REDUCTION. 


117 


MENTAL  PROBLEMS. 

1.  How  many  half-pint  bottles  can  be  filled  with 
gallons  of  sweet  oil? 

2.  A  boy  bought  f  of  a  bushel  of  chestnuts  for  $2,  and 
sold  them  at  10  cents  a  quart:  how  much  did  he  gain? 

3.  If  a  workman  can  do  a  job  of  work  in  120  hours, 
how  many  days  will  it  take  him  if  he  work  8  hours  a  day? 

4.  How  much  will  f  of  a  cwt.  of  sugar  cost,  at  16f 
cents  a  pound? 

5.  If  a  man  spend  ^  of  each  day  in  sleep,  how  many 
hours  will  he  sleep  in  the  last  three  months  of  the  year? 

6.  If  a  man  walk  10  hours  a  day,  at  the  rate  of  3.3 
miles  an  hour,  how  far  will  he  walk  in  6  days? 

7.  How  many  square  inches  in  the  surface  of  a  brick  8 
inches  long,  4  inches  wide,  and  2  inches  thick? 

8.  How  many  square  feet  in  a  board  12.6  ft.  long  and 
8  inches  wide? 

9.  How  many  solid  feet  in  a  plank  16  feet  long,  1^  feet 
wide,  and  4  inches  thick? 

10.  A  man  paid  $36  for  a  stack  of  hay  containing  4j 
tons,  and  sold  it  at  50  cents  a  hundred :  how  much  did  he 
gain  ? 


WRITTEN  PROBLEMS. 

11.  How  many  yards  of  carpeting,  f  of  a  yard  wide,  will 
carpet  a  room  27  feet  long  and  21^  feet  wide? 

12.  How  many  acres  in  a  street  2^  miles  long  and  5  rods 
wide  ? 

13.  What  would  be  the  cost  of  a  township  of  land  6 
miles  square,  at  $10.50  an  acre? 

14.  A  rectangular  field  is  60  rods  long  and  37^  rods 
wide :  how  many  boards,  each  12  feet  long,  will  inclose  it 
with  a  fence  5  boards  high? 

15.  At  $5.62^-  a  cord,  what  will  be  the  cost  of  a  pile  of 
wood  85  ft.  6  in.  long,  6  ft.  4  in.  high,  and  4  ft.  wido? 


118 


COMPLETE  ARITHMETIC. 


16.  How  many  bricks,  4  by  8  in.,  will  it  take  to  pave  a 
walk  16  feet  wide  and  6^-  rods  long? 

17.  How  many  gold  rings,  each  weighing  3.2  pwt.,  can 
be  made  from  a  bar  of  gold  weighing  .75  of  a  pound? 

18.  An  octavo  book  contains  480  pages  :  how  many  reams 
of  paper  will  it  take  to  print  an  edition  of  1200  copies, 
making  no  allowance  for  waste? 

19.  How  many  perches  of  masonry  in  the  wall  of  a  cellar 
45  feet  long,  34  feet  wide,  8  feet  high,  and  2 J  feet  thick  ? 

Note. — In  measuring  walls  of  cellars  and  buildings,  masons  take 
the  distance  round  the  outside  of  the  walls  (the  girth)  for  the  length, 
thus  measuring  each  corner  twice. 

20.  How  many  perches  of  masonry  in  the  walls  of  a  fort 
120  feet  square,  the  walls  being  33J  feet  high  and,  on  an 
average,  11  feet  thick? 

21.  What  will  it  cost  to  excavate  a  cellar  40  ft.  long, 
21  ft.  6  in.  wide,  and  4  ft.  deep,  at  $1.75  a  cubic  yard? 

22.  A  bin  is  8  ft.  long,  3^-  ft.  wide,  and  4  ft.  deep :  how 
many  bushels  of  grain  (2150|  cu.  in.)  will  it  hold? 

23.  A  circular  park  is  165  yards  in  diameter:  how  many 
acres  does  it  contain? 

24.  How  many  cubic  feet  in  the  capacity  of  a  round 
well  3J  ft.  in  diameter  and  20  ft.  deep? 

25.  A  round  cistern  is  5  ft.  in  diameter  and  6  ft.  4  in. 
deep:  how  many  gallons  of  water  will  it  hold? 

26.  A  congressional  township  is  6  miles  square,  and  is 
divided  into  36  sections:  how  many  acres  in  a  section? 

27.  A  tract  of  land  is  4  miles  long  and  2^  miles  wide: 
how  many  sections  does  it  contain?  How  many  acres  does 
it  contain? 

28.  A  speculator  bought  3^  sections  of  land  at  $4.50  an 
acre,  and  sold  them  at  $6.25  an  acre:  how  much  did  he 

gain?'  n 

29.  A  man  sold  a  farm  containing  a  quarter  of  a  section 
of  land,  for  $3280:  what  did  he  receive  per  acre? 


THE  METRIC  SYSTEM. 


119 


THE  METRIC  SYSTEM. 

178.  The  Metric  System  is  a  system  of  weights 
and  measures  expressed  on  the  decimal  scale. 

The  system  was  first  adopted  by  France,  and  it  is  now  in  general 
use  in  nearly  all  the  countries  of  Europe.  The  use  of  the  system  in 
the  United  States  was  legalized  by  Congress  in  1866,  and  it  is  em¬ 
ployed,  to  some  extent,  in  several  departments  of  the  government 
service.  It  has  long  been  used  by  the  Coast  Survey. 

The  convenience  and  accuracy  of  the  system  have  secured  its 
very  general  adoption  in  the  sciences  and  in  the  arts. 

179.  The  Meter  is  the  primary  unit  of  the  system.  It 
is  the  length  of  a  bar  of  metal  kept  at  Paris  as  a  standard. 

The  meter  was  intended  to  be  the  ten-millionth  part  of  the  distance 
from  the  equator  to  the  north  pole,  but  subsequent  measurements  of 
this  quadrant  show  that  its  length  is  a  little  more  than  ten  million 
meters. 

A  standard  meter,  copied  from  the  one  at  Paris,  is  kept  by  each 
nation  that  has  adopted  the  metric  system.  The  standard  meter  of 
the  United  States  is  kept  at  Washington.  Its  length  is  39.37 
inches. 

The  Titer  ( le'-ter )  is  the  unit  of  the  measures  of  ca¬ 
pacity.  It  is  the  thousandth  part  of  a  cubic  meter. 

The  Gram  is  the  unit  of  weights.  It  is  the  weight  of 
the  thousandth  part  of  a  liter  of  water  at  its  greatest 
density. 

180.  The  meter,  liter,  and  gram  are  each  multiplied  by 
10,  100,  1000,  and  10000,  giving  multiple  units,  and  they 
are  also  each  divided  by  10,  100,  1000,  giving  the  decimal 
subdivisions  of  tenths,  hundredths,  thousandths,  etc. 

181.  The  multiples  are  named  by  prefixing  to  the  name 
of  the  primary  unit,  or  base,  the  Greek  numerals,  Veka 
(10),  Hekto  (100),  Kilo  (1000),  and  Myria  (10000);  and 
the  subdivisions  are  named  by  prefixing  the  Latin  words, 
Deci  (10th),  Centi  (100th),  and  Midi  (1000th). 


120 


COMPLETE  ARITHMETIC. 


METRIC  TABLES. 

182. — I.  Measures  of  Length. 


The  Unit  is  a  Meter  =  39.37  inches. 

Denominations  Abbreviations.  Values. 


Myriameter  .  . 

.  .  Mm. 

= 

10000  meters. 

Kilometer  .  . 

.  .  Km. 

= 

1000  “ 

Hektometer .  . 

.  .  Hm. 

= 

100  “ 

Dekameter  .  . 

.  .  Dm. 

= 

10  “ 

Meter  .... 

= 

1  meter. 

Decimeter  .  . 

.  .  dm. 

.1  “ 

Centimeter  .  . 

.  .  cm. 

= 

.01  “ 

Millimeter  .  . 

.  .  mm. 

zzz 

.001  “ 

Decimal  Scale. 


00000.000 


Ten  units  of  anv  denomination  of  the  above 
table  equal  one  unit  of  the  next  higher  de¬ 
nomination,  and,  hence,  the  successive  de¬ 
nominations  correspond  to  successive  orders 
of  figures  in  the  decimal  system :  the  meter 
denoting  units;  the  dekameter,  tens,  etc. 

The  correspondence  between  the  metric 
denominations  and  those  of  United  States 
Money  is  also  noticeable.  The  millimeter  cor¬ 
responds  to  mills ;  the  centimeter  to  cents;  the 
decimeter  to  dimes  ;  the  meter  to  dollars ,  etc. 


The  above  diagram  shows  that  a  decimeter  is  a  little  less  than  four 
inches,  and  that  a  centimeter  is  a  little  more  than  f  of  an  inch. 


Note. — The  spelling,  pronunciation,  and  abbreviations  employed 
are  those  recommended  by  the  Metric  Bureau,  Boston,  and  the 
American  Metrological  Society.  The  equivalents  of  the  metric 
units  are  those  legalized  by  Congress. 


THE  METRIC  SYSTEM. 


121 


183. — II.  Measures  of  Surface. 


The  Unit  is  an  Ar,  a  Square  Dekameter  =  3.95  sq.  rd. 


Denominations. 


Abbreviations.  Values. 


Hektar . Ha.  =  10.000  sq.  m. 

At  (are) . a.  =  100  sq.  m. 

Centar . ca.  =  1  sq.  m. 


Decimal  Scale. 


a 


Since  100  units  of  each  denomination  in  the  above 
table  equal  one  of  the  next  higher,  each  occupies  two 
orders  of  figures.  The  centars  correspond,  in  this 
respect,  to  cents,  which  occupy  two  places. 

The  above  table  is  used  in  measuring  land. 

The  primary  unit  for  the  measuring  of  small  surfaces  is  a  square 
meter. 


S-i 

a 

24 

<D 

M 


U 


00  0.00 


Note. — Centar  is  also  written  centar e. 


184. — III.  Measures  of  Capacity. 

The  Unit  is  a  Liter,  a  cubic  Decimeter  =  .908  dry  qt.  =  1.056  liq.  qt. 


Denominations.  Abbreviations.  Values. 

Kiloliter . Kl.  =  1000  liters. 

Hektoliter . HI.  =  100  liters. 

Dekaliter . Dl.  =  10  liters. 

Liter . 1.  =  1  liter. 


Deciliter . dl.  =  .1  liter. 

Centiliter . cl.  =  .01  liter. 

Miililiter . ml.  =  .001  liter. 

The  kiloliter  equals  a  cubic  meter,  the  liter  a  cubic  decimeter,  and 

the  millimeter  a  cubic  centimeter. 

The  kiloliter  is  called  a  ster  (st.),  and  is  the  principal  measure  of 
wood,  stone,  etc.  One  tenth  of  a  ster  is  a  decister  (dst.)  and  10  sters 
are  a  dekaster  (Dst.). 

The  liter  is  used  in  measuring  liquids,  and  the  hektoliter  in  meas¬ 
uring  grains. 

Note.— When  the  three  dimensions  of  a  regular  solid  are  expressed 
in  decimeters,  their  products  will  be  the  contents  in  liters. 

C  Ar.— 11. 


122 


COMPLETE  ARITHMETIC. 


185. — IV.  Weights. 

The  Unit  is  a  Gram  =  15.432  gr.  Troy  =  .035  oz.  Av. 


Denominations.  Abbreviations.  Values. 

Ton . T.  =  1000000  grams. 

Myriagram  ....  Mg.  =  10000  grams. 

Kilogram,  or  Kilo  .  .  K.  =  1000  grams. 

Hektogram  ....  Hg.  =  100  grams. 

Dekagram . Dg.  =  10  grams. 

Gram  .....  g.  =  1  gram. 

Decigram . dg.  =  .1  gram. 

Centigram . eg.  =  .01  gram. 

Milligram . mg.  =  .001  gram. 


A  ton  (1000 Kg)  equals  the  weight  of  a  cubic  meter  of  water  at  its 
greatest  density;  a  kilogram  equals  a  liter  of  water;  the  gram,  a 
cubic  centimeter  of  water;  and  the  milligram,  a  cubic  millimeter  of 
water.  The  kilogram,  called  for  brevity  kilo,  is  the  ordinary  weight 
of  commerce. 

Note. — The  nickel  five-cent  piece  weighs  5  grams,  and  a  silver 
half-dollar  12^  grams.  The  weight  of  a  letter  for  single  postage 
(2  cents)  must  not  exceed  the  weight  of  3  nickels  (15  grams).  The 
nickel  is  2  centimeters  in  diameter. 


186. — V.  Metric  Equivalents. 


Long  Measure. 

An  inch  =  .0254  meter. 

A  foot  =  .3048  meter. 

A  yard  =  .9144  meter. 

A  rod  ==  5.029  meters. 

A  mile  =  1.6094  kilometers. 

Square  Measure. 

A  sq.  inch  =  .000645  sq.  m. 
A  sq.  foot  =  .0929  sq.  m. 

A  sq.  yard  =  .8362  sq.  m. 

A  sq.  rod.  =  .2529  ar. 

An  acre  =  .4047  hektar. 

A  sq.  mile  =  259  hektars. 


Cubic  Measure. 

A  cu.  inch  =  .0164  liter. 

A  cu.  foot  =  .2832  hektoliter. 

A  cu.  yard  =  .7646  ster. 

A  cord  =*  3.6245  sters. 

Weight. 

A  grain  =  .0648  gram. 

A  pound  Av.  =  .4536  kilogram. 
A  pound  Troy  =  .373  kilogram. 
A  ton  =  .907  ton. 


A  gallon  =  3.786  liters. 

A  bushel  =  .3524  hektolitei. 


THE  METRIC  SYSTEM. 


123 


MENTAL  PROBLEMS. 

1.  How  many  meters  in  a  dekameter?  In  a  hektometer? 
A  kilometer?  A  myriameter? 

2.  What  part  of  a  meter  is  a  decimeter  ?  A  centimeter  ? 
A  millimeter? 

3.  Name  the  metric  units  of  length,  in  order,  from  the 
smallest  to  the  greatest. 

4.  How  many  liters  in  a  hektoliter?  In  a  dekaliter?  A 
kiloliter  ? 

5.  Name  the  metric  units  of  capacity  from  the  smallest 
to  the  greatest. 

6.  What  part  of  a  gram  is  a  centigram?  A  decigram? 
A  milligram? 

7.  How  many  meters  in  5  dekameters  ?  44  dekameters  ? 

225  dekameters?  34.6  dekameters? 

8.  How  many  liters  in  3  hektoliters?  37  hektoliters? 
22.5  hektoliters?  7.45  hektoliters? 

9.  How  many  grams  in  8  kilograms?  24  kilograms? 
3.25  kilograms?  .456  kilogram? 

10.  What  decimal  part  of  a  meter  is  a  centimeter?  15 
centimeters?  72  centimeters? 

11.  What  decimal  part  of  a  gram  is  a  milligram?  24 
milligrams?  245  milligrams? 

12.  When  the  metric  units  are  expressed  on  the  decimal 
scale,  which  order,  from  the  decimal  point,  is  the  decimeter? 
The  millimeter?  The  centimeter? 

13.  Which  order,  from  the  decimal  point,  is  the  deka¬ 
liter?  The  kiloliter?  The  hektoliter? 

14.  Read  the  several  orders  in  324.56  meters  as  metric 
units. 

Arts.  3  hektometers  2  dekameters  4  meters  5  decimeters  and  6 
centimeters. 

15.  Read  the  several  orders  in  504.046  grams  as  metric 
units. 

16.  Read  4080.57  liters  in  metric  units. 


124 


COMPLETE  ARITHMETIC. 


WRITTEN  PROBLEMS. 

17.  Write  5Ke  7IIS  6Dg  5  s  and  6cg  on  the  decimal  scale 

as  grams.  ,  Ans.  5765.06  grams. 

18.  Write  6m  4D1  31  and  5dl  on  the  decimal  scale  as 
liters. 

19.  How  many  meters  in  6Km  7Dm  and  5dm? 

*20.  How  many  grams  in  5Ks  3Dg  4g  6dg?  In  7Kg  8Hg 

3s  5<is  7 cg 9  In  6 Kg  40g  and  8cg? 

21.  Reduce  234.56  hektograms  to  grams. 

Process  :  234.56X100  =  23456.  Ans.  23456  grams. 

22.  Reduce  345.8  centigrams  to  grams. 

Process:  345.8  -s-  100  =  3.458.  Ans.  3.458  grams. 

23.  Reduce  45.06  kiloliters  to  liters. 

24.  Reduce  35.4  hektoliters  to  liters. 

25.  Reduce  84.5  ars  to  square  meters. 

26.  Reduce  132.4  centimeters  to  meters. 

27.  Reduce  24000  millimeters  to  meters. 

28.  Reduce  434.5  centiliters  to  liters. 

29.  Reduce  3.225  quintals  to  grams. 

30.  Reduce  746.35  dekagrams  to  kilos. 

31.  How  many  yards  in  220  meters. 

Process:  39.37  in.  X  220  -s-  12  =  3  =  240.59  +.  Ans.  240.59  yd. 

32.  How  many  miles  in  44.5  kilometers? 

33.  How  many  inches  in  24  centimeters? 

34.  How  many  bushels  in  250  hektoliters  of  wheat? 

35.  How  many  gallons  in  37J  liters  of  sirup? 

36.  How  many  pounds  of  butter  in  150  kilos  ? 

37.  How  many  liters  in  35  cubic  feet? 

38.  How  many  sters  in  20  cords  of  wood? 

39.  How  many  ars  in  -J  of  an  acre? 

40.  How  many  meters  in  1760  feet? 

*41.  How  many  square  meters  in  a  floor  9.25  m  long  and 
6.8  m  wide?  12.45m  loDg  and  8.6m  wide? 


COMPOUND  NUMBERS. 


125 


SECTION  XIII. 

COMPOUND  NUMBERS. 

ADDITION  AND  SUBTRACTION. 

1.  What  is  the  sum  of  7  cwt.  44  lb.  6  oz.  11.5  dr.;  12  cwt. 
13  lb.  7.6  dr.;  23  cwt.  56  lb.  12  oz. ;  and  27  cwt.  14  oz. 
8.4  dr.  ? 

Since  only  like  numbers  can  be  added, 
write  the  numbers  of  the  same  denomina¬ 
tion  in  the  same  columns.  The  sum  of  the 
drams  is  27.5  dr.  =  1  oz.  11.5  dr.  Write 
the  11.5  dr.  under  drams,  and  add  the  1  oz. 
with  the  ounces.  Proceed  in  like  manner 
until  the  numbers  of  the  several  denomi¬ 
nations  are  added. 

2.  Add  16  mi.  7  fur.  27  rd.  3  yd.  2  ft.  8^  in.;  13  mi. 
4  fur.  5  yd.  1  ft.  7-f  in. ;  27  mi.  35  rd.  4  yd.  5J  in. ;  and 
6  fur.  24  rd.  3  yd.  2  ft. 

3.  Add  13  w.  6  d.  13  h.  48  min.;  8  w.  13  h.  51  min.  37  sec.; 
12 w.  5d.  22 h.  16  min.  44  sec.;  1  w.  10 h.  15  min.;  and  Id. 
10  h.  26  sec. 

4.  Add  24  lb.  10  oz.  17  pwt.  22  gr.;  16  lb.  19  pwt.;  10  oz. 
15  pwt.  21  gr.;  45  lb.  9  oz.  18  gr.;  and  13  lb.  11  oz.  18  pwt. 
23  gr. 

5.  Add  15  bu.  3  pk.  7  qt. ;  27  bu.  5  qt.  1  pt. ;  8  bu. 
2  pk.  1  pt. ;  47  bu.  3  pk. ;  12  bu.  2  pk.  1  qt.  1  pt. ;  and 
19  bu.  1  pk.  3  qt. 

6.  Add  16°  32'  43";  28°  47'  53";  25°  53";  4  s.  48'  48"; 
Us.  16°  36'  59";  and  5s.  18°  7'  8". 

7.  How  many  cords  of  wood  in  three  piles,  the  first  being 
23  ft.  long,  4  ft.  wide,  and  7  ft.  high;  the  second,  28  ft. 
long,  4  ft.  wide,  and  6^  ft.  high;  and  the  third,  17  ft.  long, 
8  ft.  wide,  and  7J  ft.  high? 


Process. 


cwt. 

lb. 

oz. 

dr. 

7 

44 

6 

11.5 

12 

13 

0 

7.6 

23 

56 

12 

0. 

27 

00 

14 

8.4 

10 

15 

1 

11.5 

12(5 


f 


COMPLETE  ARITHMETIC. 


8.  From  13  rd.  3  yd.  1  ft.  6.4  in.  take  9  rd.  4  yd.  11.5  in. 


Process. 

rd.  yd.  ft.  in. 

13  3  1  6.4 

9  4  0  11.5 

3  41  0  6.9 

1  =  1  6 

3  4  2  .9 


Write  the  subtrahend  under  the  minuend, 
placing  the  numbers  of  the  several  denomina¬ 
tions  in  columns,  as  in  compound  addition. 
Since  11.5  in.  is  greater  than  6.4  in.,  add  12  in. 
to  6.4  in.  and  then  subtract.  To  balance  the 
12  in.  added  to  the  minuend,  add  1  ft.  (12  in.) 
to  the  subtrahend  (Art.  30,  Pr.  3),  or,  if  pre¬ 
ferred,  subtract  1  ft.  from  the  minuend. 


Proceed  in  like  manner  until  the  difference  between  the  numbers 
of  the  several  denominations  is  found.  Reduce  the  ^  yd.  to  feet  and 
inches,  and  add  the  result  to  the  0  ft.  6.9  in.  of  the  remainder. 


9.  From  30  mi.  6  fur.  14  rd.  3  yd.  1  ft.  4  in.  take  25  mi. 
36  rd.  4  yd.  2  ft.  10  in. 

10.  From  33  rd.  1  yd.  2  ft.  11  in.  take  16  rd.  3  yd.  8  in. 

11.  From  104°  11'  20"  take  83°  43'  36". 

12.  Boston  is  71°  4'  9"  W.  longitude,  and  San  Francisco 
is  122°  26'  15"  W.  longitude :  what  is  their  difference  in 
longitude  ? 

13.  From  the  sum  of  245  A.  2  R.  27  P.  and  187  A.  3  R. 
34  P.  take  their  difference. 

14.  A  note  was  given  July  23,  1863,  and  it  was  paid  Nov. 
16,  1868 :  how  long  did  it  run  ? 

15.  A  man  wras  born  Sept.  12,  1827,  and  his  eldest  son 
was  born  Apr.  6,  1855:  what  is  the  difference  in  their  ages? 

16.  Baltimore  is  situated  76°  37'  W.,  and  Vienna  16° 
23'  E. :  what  is  their  difference  in  longitude  ? 

17.  A  ship  in  latitude  37°  20'  north,  sails  15°  45'  south; 
then  12°  36'  north;  then  18°  40'  south:  what  is  her  latitude ? 


DEFINITIONS  AND  RULES. 

187.  A  Compound  Number  is  a  number  composed 
of  units  of  several  denominations.  (Art.  173.) 

188.  Compound  numbers  are  of  the  same  hind  when  their 
corresponding  terms  denote  units  of  the  same  denomination; 
as,  3  bu.  2  pk.,  and  6  bu.  3  pL  5  qt. 


COMPOUND  NUMBERS. 


127 


189.  Compound  Addition  is  the  process  of  finding 
the  sum  of  two  or  more  compound  numbers  of  the  same 
kind. 

190.  Rule. — To  add  compound  numbers, 

1.  Write  the  compound  numbers  to  be  added  so  that  units  of 
the  same  denomination  shall  stand  in  the  same  column . 

2.  Add  the  column  of  the  lowest  denomination,  and  divide 
the  sum  by  the  number  of  units  of  that  denomination ,  which 
equals  a  unit  of  the  next  higher  denomination ;  write  the  re¬ 
mainder  under  the  column  added,  and  add  the  quotient  ivith 
the  next  column. 

3.  In  like  manner  add  the  remaining  columns,  writing  the 
sum  of  the  highest  column  under  it. 

Note.- — In  both  simple  and  compound  addition,  the  sum  of  each 
column  is  divided  by  the  number  of  units  of  that  denomination,  which 
equals  one  of  the  next  higher  denomination.  In  simple  addition  this 
divisor  is  10 ;  in  compound  addition  it  is  a  varying  number,  since 
the  several  denominations  are  expressed  on  a  varying  scale. 

191.  Compound  Subtraction  is  the  process  of 
finding  the  difference  between  two  compound  numbers  of 
the  same  kind. 

192.  Rule. — To  subtract  one  compound  number  from 
another, 

1.  Write  the  subtrahend  under  the  minuend,  placing  terms 
of  the  same  denomination  in  the  same  column. 

2.  Beginning  at  the  right,  subtract  each  successive  term  of  the 
subtrahend  from  the  corresponding  term  of  the  minuend,  and 
write  the  difference  beneath. 

3.  If  any  term  of  the  subtrahend  be  greater  than  the  corre¬ 
sponding  term  of  the  minuend,  add  to  the  term  of  the  minuend 
as  many  units  of  that  denomination  as  equal  one  of  the  next 
higher,  and  from  the  sum  subtract  the  term  of  the  subtrahend, 
writing  the  difference  beneath. 

4.  Add  one  to  the  next  term  of  the  subtrahend,  and  proceed 
as  before. 

Note. — Instead  of  adding  one  to  the  next  term  of  the  subtrahend, 
one  may  be  subtracted  from  the  next  term  of  the  minuend. 


128 


COMPLETE  ARITHMETIC. 


MULTIPLICATION  AND  DIVISION. 


1.  Multiply  15  rd.  3  yd.  1  ft.  7  in.  by  11. 

Process. 


Since  the  value  of  the  units  of 
the  successive  denominations  increases 
from  right  to  left,  begin  at  the  right 
hand.  11  times  7  in.  =-  77  in.  =  6  ft. 
5  in.  Write  the  5  in.  under  inches, 
and  reserve  the  6  ft.  to  add  with  the  product  of  feet.  Proceed  in 
like  manner  until  the  numbers  of  the  several  denominations  are 
multiplied. 


15  rd.  3  yd.  1  ft.  7  in. 
_ 11 

4  fur.  11  rd.  5  yd.  2  ft.  5  in. 


2.  If  a  man  can  build  7  rd.  11  ft.  6  in.  of  fence  in  a 
day,  how  much  can  15  men  build? 

3.  How  many  bushels  of  wheat  in  18  bins,  each  con¬ 
taining  124  bu.  3  pk.  5  qt.  ? 

4.  How  much  hay  in  13  stacks,  each  containing  4  T. 
13  cwt.  56  lb.  ? 

5.  What  is  the  weight  of  12  silver  spoons,  each  weigh¬ 
ing  2  oz.  13  pwt.  14  gr.  ? 

6.  Divide  19  mi.  4  fur.  20  rd.  2  yd.  9  in.  by  7. 

Process.  Since  the  value  of  the 

7 )  19  mi.  4  fur.  20  rd.  2  yd.  0  ft.  9  in.  units  of  the  successive  de- 

2  mi.  6  fur.  14  rd.  1  yd.  2  ft.  in.  nominations  decreases  from 

left  to  right,  begin  at  the 
left  hand.  \  of  19  mi.  =  2  mi.  with  5  mi.  remaining.  Write  the 

2  mi.  under  miles,  and  reduce  the  5  mi.  to  furlongs,  and  add  the 

4  fur.  which  gives  44  fur.  \  of  44  fur.  =  6  fur.  with  2  fur.  remain¬ 
ing.  Reduce  the  2  fur.  to  rods,  add  the  20  rd.,  take  4  of  the  result, 
and  proceed  in  like  manner  until  the  numbers  of  all  the  denomina¬ 
tions  are  divided. 


7.  Divide  27  mi.  3  fur.  25  rd.  12  ft.  6  in.  by  12. 

8.  A  ship  sailed  39°  12'  40"  in  21  days:  how  many  de¬ 
grees  did  it  average  each  day? 

9.  If  15  equal  bars  of  silver  contain  24  lb.  8  oz.  16  pwt., 
what  is  the  weight  of  each  bar  ? 

10.  If  12  equal  bins  hold  430  bu.  2  pk.  of  wheat,  how 
much  wheat  is  there  in  each  bin  ? 


COMPOUND  NUMBERS. 


129 


11.  From  13  w.  5d.  18  h.  40  min.  take  7  w.  23 h.  45  min., 
and  divide  the  difference  by  15. 

12.  Add  4  fur.  23  rd.  3  yd.  2  ft.  and  7  fur.  16  rd.  1  ft., 
and  divide  the  sum  by  22. 

13.  From  the  sum  of  56  lb.  13  oz.  9  dr.  and  47  lb.  15  oz. 
15  dr.  take  their  difference,  and  divide  the  result  by  9. 

14.  How  many  rings,  each  weighing  4  pwt.  15  gr.,  can  be 
made  from  a  bar  of  gold  weighing  1  lb.  10  oz.  ? 

Suggestion. — Reduce  both  divisor  and  dividend  to  the  same  de¬ 
nomination. 

15.  How  many  kegs,  each  containing  5  gal.  1  qt.,  can  be 
filled  from  a  cask  holding  63  gal.  ? 

16.  How  many  rotations  will  a  wheel  12  ft.  6  in.  in  cir¬ 
cumference  make  in  rolling  f  of  a  mile  ? 

17.  How  many  lengths  of  fence,  each  11  ft.  6  in.,  will 
inclose  a  square  field  each  side  of  which  is  20  rd.  5  yd.  ? 

18.  How  many  barrels,  each  holding  2  bu.  3  pk.,  will 
hold  132  bushels  of  apples? 

19.  How  many  axes,  each  weighing  3  lb.  3  oz.,  can  be 
made  from  a  ton  of  iron? 

20.  How  many  steps,  2  ft.  6  in.  each,  will  a  man  take 
in  walking  round  a  field  45  rods  square? 

21.  The  length  of  a  solar  year  is  365  d.  5  h.  48  min. 
48  sec. :  how  much  time  is  of  a  solar  year  ? 

DEFINITIONS  AND  RULES. 

193.  Compound  Multiplication  is  the  process  of 
taking  a  compound  number  a  given  number  of  times.  The 
multiplier  is  always  an  abstract  number. 

194.  Rule. — To  multiply  a  compound  number, 

1.  Write  the  multiplier  under  the  loiuest  denomination  of  the 
multiplicand. 

2.  Beginning  at  the  right ,  multiply  each  term  of  the  multi¬ 
plicand  in  order ,  and  reduce  each  product  to  the  next  higher 


130 


COMPLETE  ARITHMETIC. 


denomination ,  writing  the  remainder  under  the  term  multiplied , 
and  adding  the  quotient  to  the  next  product 

Note. — In  both  simple  and  compound  multiplication,  the  successive 
products  are  each  divided  by  the  number  of  units  of  their  denomination 
which  equal  one  of  the  next  higher  denomination. 

195.  Compound  Division  is  the  process  of  dividing 
a  compound  number  into  equal  parts. 

198.  Rules. — I.  To  divide  a  compound  number, 

1.  Write  the  divisor  at  the  left  of  the  dividend,  as  in  simple 
division. 

2.  Beginning  at  the  left,  divide  each  term  of  the  dividend  in 
order,  and  write  the  quotient  under  the  term  divided. 

3.  If  the  division  of  any  term  give  a  remainder,  reduce  the 
remainder  to  the  next  lower  denomination,  to  the  result  add  the 
number  of  that  denomination  in  the  dividend,  and  then  divide 
as  above. 

Note. — When  the  divisor  is  a  large  number,  the  successive  terms 
of  the  quotient  may  be  written  at  the  right  of  the  dividend,  as  in 
long  division. 

II.  To  divide  a  compound  number  by  another  of  the 
same  kind,  Reduce  both  compound  numbers  to  the  same  de¬ 
nomination,  and  then  divide  as  in  simple  division. 

Note. — This  is  not  properly  compound  division,  since  the  com¬ 
pound  numbers  are  reduced  to  simple  numbers  before  dividing. 


LONGITUDE  AND  TIME. 

197.  Longitude  is  distance  east  or  west  from  a  given 
meridian.  It  is  measured  in  degrees,  minutes,  and  seconds. 
Thus,  15°  24'  40"  east  longitude  denotes  a  position  15°  24' 
40"  east  of  the  meridian  from  which  longitude  is  reckoned. 

Since  every  circle  is  divided  into  360  degrees,  the  length  of  a 
degree  depends  upon  the  size  of  the  circle  of  Avhich  it  is  a  part. 

The  length  of  a  degree  of  longitude  depends  upon  the  latitude  of  the 
parallel  on  which  it  is  measured.  It  is  greatest  at  the  equator,  where 
it  is  G9£  miles,  nearly;  and  least  at  the  poles,  where  it  is  nothing. 


LONGITUDE  AND  TIME. 


131 


198.  The  earth  rotates  on  its  axis  from  west  to  east 
once  every  twenty  -  four 
hours,  and  the  illumi¬ 
nated  space  between  any 
two  meridians  passes  un¬ 
der  the  sun’s  rays  in  the 
same  length  of  time.  A 
degree  of  surface  at  the 
equator  passes  under  the 
sun’s  rays  in  the  same 
time  as  a  degree  at  any 

latitude  between  the  equator  and  the  polar  circle. 


199.  When  the  vertical  rays  of  the  sun  are  on  the 
meridian  of  any  place,  it  is  noon,  or  12  o’clock,  at  that 
place;  and  since  the  sun’s  rays  pass  over  the  earth’s  sur¬ 
face  from  east  to  west ,  it  is  after  noon  at  all  places  east  of 
this  meridian,  and  before  noon  at  all  places  west  of  it. 
When  it  is  noon  at  Cincinnati,  it  is  after  noon  at  New 
York,  and  before  noon  at  St.  Louis. 


If  24  clocks  were  placed 
15°  apart  on  any  parallel  of 
latitude  between  the  polar 
circles,  the  difference  in  time 
between  any  two  consecutive 
clocks  would  be  one  hour; 
and  the  24  clocks  would  to¬ 
gether  represent  every  hour 
of  the  day.  The  figures  in 
the  diagram  represent  the 
location  of  the  clocks  (15° 
apart),  and  also  the  hour  of 
meridian. 


MENTAL  PROBLEMS. 

1.  The  earth  rotates  on  its  axis  once  every  24  hours: 
what  part  of  a  rotation  does  it  make  in  1  hour? 

2.  How  many  degrees  of  the  earth’s  surface  pass  under 
the  sun’s  ravs  in  24  hours?  In  1  hour? 

V 


132 


COMPLETE  ARITHMETIC. 


3.  How  many  degrees  of  longitude  make  a  difference  of 
1  hour  in  time  ? 

4.  When  it  is  noon  at  Washington,  what  is  the  hour 
of  day  15°  east  of  Washington?  15°  west  of  Washington? 

5.  When  it  is  6  o’clock  at  Boston,  what  is  the  hour  of 
day  30°  east  of  Boston?  45°  west  of  Boston? 

6.  If  15°  of  longitude  give  a  difference  of  1  hour  in 
time,  how  much  longitude  will  give  a  difference  of  1  min¬ 
ute  in  time? 

7.  When  it  is  4  o’clock  at  Cincinnati,  what  is  the  time 
15'  east  of  Cincinnati?  45'  west  of  Cincinnati? 

8.  When  it  is  9  o’clock  at  Chicago,  what  is  the  time 
15°  15'  east  of  Chicago?  15°  45'  wTest  of  Chicago? 

9.  If  15'  difference  in  longitude  gives  a  difference  of  1 
minute  in  time,  what  difference  in  longitude  will  give  a 
difference  of  1  second  in  time  ? 

10.  What  difference  in  longitude  gives  a  difference  of  1 

hour  in  time  ?  1  minute  ?  1  second  ? 

11.  The  difference  in  time  between  two  cities  is  2  hours : 
what  is  the  difference  in  their  longitude?  Which  has  the 
earlier  time? 

12.  The  difference  in  time  between  New  York  and  St.  Louis 
is  1  h.  2^  min. :  what  is  the  difference  in  their  longitude  ? 

13.  A  gentleman  left  Boston  and  traveled  until  his  watch 
was  1  h.  3  min.  too  slow :  how  far  had  he  traveled,  and  in 
which  direction? 

14.  Two  captains  observed  an  eclipse  of  the  moon,  one 
seeing  it  at  9  P.  M.  and  the  other  at  11 J  P.  M. :  what  was 
the  difference  in  their  longitude? 

WRITTEN  PROBLEMS. 

15.  The  difference  in  longitude  between  two  places  is  31° 
45'  30":  what  is  the  difference  of  time? 

Process. 

15  )  31°  45/  30"  Divide  by  15  as  in  compound  division. 

2  h.  7  min.  2  sec. 


LONGITUDE  AND  TIME. 


133 


16.  The  difference  in  longitude  between  two  cities  is  5° 
31':  what  is  the  difference  in  time? 

17.  The  longitude  of  Cincinnati  is  84°  26'  W.,  and  San 
Francisco  is  122°  26'  15"  W. :  wdien  it  is  noon  at  Cincin¬ 
nati  what  is  the  time  at  San  Francisco? 

18.  Philadelphia  is  75°  10'  W.  :  when  it  is  noon  at  San 
Francisco  what  is  the  time  at  Philadelphia? 

19.  Boston  is  71°  4'  9"  W. :  when  it  is  7  P.  M.  at  Boston 
what  is  the  time  at  Cincinnati  ?  At  San  Francisco  ? 

20.  Berlin  is  13°  23'  53"  E. :  when  it  is  noon  at  Boston 
what  is  the  time  at  Berlin? 

21.  The  difference  in  time  between  two  cities  is  1  h. 
35  min.  12  sec.:  wThat  is  their  difference  in  longitude? 

Process. 

1  h.  35  min.  12  sec.  Multiply  by  15  as  in  compound  multipli- 

_ _ 15  cation. 

23°  48'  0",  Ans. 

22.  The  difference  in  time  in  the  observations  of  an 
eclipse,  on  two  vessels  at  sea,  is  2  h.  15  min.  10  sec. :  what 
is  their  difference  in  longitude? 

23.  An  eclipse  was  observed  at  New  York,  74°  W.,  at 
9.30  P.  M.,  and  the  time  of  its  observation  on  a  vessel  in 
the  Atlantic  Ocean,  was  11.45  P.  M. :  what  was  the  longi¬ 
tude  of  the  vessel  ? 

24.  The  difference  in  time  between  the  chronometers  of 
two  observatories  is  45  min.  30  sec.,  and  the  longitude  of 
the  observatory  having  the  faster  time  is  85°  40'  W. : 
what  is  the  longitude  of  the  other  observatory? 

25.  The  distance  from  Boston  to  Chicago  is  about  843 
miles,  and  a  degree  of  longitude  at  Boston  contains  about 
51  miles :  when  it  is  noon  at  Boston  what  is  the  time  at 
Chicago  ? 

26.  The  distance  from  Washington  to  St.  Louis  is  about 
714  miles,  and  a  degree  of  longitude  at  Washington  con¬ 
tains  about  54  miles :  when  it  is  9  o’clock  at  St.  Louis  what 
is  the  time  at  Washington9 


134 


COMPLETE  ARITHMETIC. 


27o  The  difference  in  the  longitude  of  two  vessels,  at  the 
time  of  the  observation  of  an  eclipse,  was  25°  30':  what 
was  their  difference  in  time? 

28.  How  much  earlier  does  the  sun  rise  at  Baltimore, 
which  is  76°  37'  W.,  than  at  Cincinnati,  84°  26'  W.  ? 

29.  How  much  later  does  the  sun  set  at  Chicago,  which 
is  87°  35'  W.,  than  at  Boston,  71°  4'  9"  W.  ? 

30.  How  much  later  does  the  sun  set  at  San  Francisco, 
which  is  122°  26'  15"  W.,  than  at  Cincinnati? 

31.  How  much  earlier  does  the  sun  rise  at  New  York, 
which  is  74°  W.,  than  at  San  Francisco? 

32.  How  much  later  does  the  sun  set  at  Boston  than  at 
Berlin,  which  is  13°  23'  53"  E.  ? 

TABLE  AND  RULES. 

200.  Table  :  15°  difference  in  long,  gives  a  difference 

of  1  h.  in  time. 

15'  difference  in  long,  gives  a  difference 
of  1  m.  in  time. 

15"  difference  in  long,  gives  a  difference 
of  1  sec.  in  time. 

201.  Rules. — 1.  To  find  the  difference  in  time  corre¬ 
sponding  to  any  difference  in  longitude,  Divide  the  difference 
in  longitude ,  expressed  in  degrees,  minutes t,  and  seconds,  by  15, 
arid  the  respective  quotients  will  he  hours,  minutes,  and  seconds 
of  time. 

2.  To  find  the  difference  in  longitude  corresponding  to 
any  difference  in  time,  Multiply  the  difference  in  time,  ex¬ 
pressed  in  hours,  minutes,  and  seconds,  hy  15,  and  the  respect¬ 
ive  products  will  be  degrees,  minutes,  and  seconds  of  longitude. 

3.  To  find  the  time  at  one  place  when  the  time  at 
another  place  and  their  difference  of  time  are  known,  When 
the  second  place  is  east  of  the  first,  add  their  difference  of 
time;  when  it  is  west  of  the  first,  subtract  their  difference 
of  time. 


PERCENTAGE. 


135 


SECTION  XIV. 

PERCENTAGE. 

NOTATION  AND  DEFINITIONS. 

202.  One  per  cent  of  a  number  is  one  hundredth  of  it ; 
two  per  cent  is  two  hundredths;  and,  generally,  any  per 
cent  of  a  number  is  so  many  hundredths  of  it. 

1.  How  many  hundredths  of  a  number  is  4  per  cent 
of  it?  7  per  cent  of  it?  15  per  cent  of  it? 

2.  How  many  hundredths  of  a  number  is  3^  per  cent 
of  it?  12^  per  cent?  33  J  per  cent? 

3.  How  many  hundredths  of  a  number  is  |  of  one  per 
cent  of  it?  f  of  one  per  cent?  ■§■  of  one  per  cent? 

4.  How  many  hundredths  of  a  number  is  115  per  cent 
of  it?  135  per  cent?  180  per  cent? 

5.  What  per  cent  of  a  number  is  .05  of  it?  .09  of  it? 
.15  of  it?  .35  of  it? 

6.  What  per  cent  of  a  number  is  .03J  of  it?  .33^  of 
it?  .00^  of  it?  .00^  of  it? 

7.  What  per  cent  of  a  number  is  of  it?  of  it? 
1.25  of  it?  1.65  of  it? 

8.  How  many  hundredths  of  a  number  is  45  %  of  it? 
110%?  125%?  170%?  215%? 

Note. — The  character  %  is  often  used  instead  of  the  words  “per 
cent.”  Thus,  15  %  denotes  15  per  cent. 

9.  What  fractional  part  of  a  number  is  10  %  of  it? 
20  %  ?  .  25  %  ?  50  %  ? 

Solution. —  50  %  of  a  number  is  T5^,  or  |  of  it. 

10.  What  fractional  part  of  a  number  is  12J  %  of  it  ? 
16|%?  33i%?  66$%? 


136 


COMPLETE  ARITHMETIC. 


11.  What  fractional  part  of  100  %  is  25  %  ?  50  %  ? 

75  %  ?  66f  %  ? 


WRITTEN  EXERCISES. 

12.  Express  decimally  1%;  3%;  6%;  7%;  8%;  9%. 

13.  Express  decimally  15%;  20%;  25%;  33%;  45%. 

14.  Express  decimally  112  %  ;  125%;  150%;  220%. 

15.  Express  decimally  6-^  %  ;  3J  %  ;  12 24f%. 

Note. — The  fractional  part  of  one  per  cent,  may  be  expressed  deci¬ 
mally,  as  thousandths,  ten-thousandths,  etc.  Thus,  6£  %  =  .06 1,  or 
.0625. 

16.  Express  decimally  7§  % ;  16f%;  10^-%;  30J%. 

17.  Express  decimally  \  % ;  |%;  |%;  f  %;  -&%. 

18.  Express  decimally  5  %  ;  x3o  %  ;  7T%  %  ;  \  %  ;  20£  %. 


DEFINITIONS. 

203.  Any  per  cent  of  a  number  or  quantity  is  so  many 
hundredths  of  it. 

204.  The  Hate  Per  Cent  is  the  fraction  denoting  the 
number  of  hundredths  taken. 

The  rate  per  cent  is  expressed  numerically  either  as  a  common 
fraction  or  as  a  decimal.  The  term  per  cent  is  a  contraction  of  the 
Latin  per  centum,  which  means  by  the  hundred. 

205.  The  Hate  is  the  number  of  hundredths  taken. 

Note. — In  this  work,  the  terms  Hate  Per  Cent  and  Pate,  are  not 
used  as  synonymous.  In  the  statement,  “  12  is  6  per  cent  of  200,” 
12  is  considered  the  percentage  (or  per  cent) ;  or  .06,  as  the 

rate  per  cent ;  and  6  as  the  rate.  The  rate  is  the  numerator  of  the 
fraction  denoting  the  rate  per  cent. 

206.  The  character  %  is  called  the  Per  Cent  Sign,  and  is 
read  per  cent. 

207.  Percentage  embraces  all  numerical  operations  in 
which  one  hundred  is  the  basis  of  computation. 


PERCENTAGE. 


137 


THE  FOUR  CASES  OF  PERCENTAGE. 

208.  Four  numbers  are  considered  in  percentage,  and  such 
is  the  relation  between  them  that,  if  any  two  of  them  are 
given,  the  other  two  may  be  found. 

These  four  numbers  are: 

1.  The  Base,  or  the  number  of  which  the  per  cent  is 
found. 

2.  The  Bate  Per  Cent,  or  the  fraction  denoting  the 
number  of  hundredths  of  the  base  taken. 

3.  The  Percentage,  or  the  part  of  the  base  corre¬ 
sponding  to  the  rate  per  cent;  also  called  the  Per  Cent. 

4.  The  Amount  or  Difference ,  or  the  number  ob¬ 
tained  by  adding  the  per  cent  to,  or  subtracting  it  from, 
the  base. 

Case  I. 

The  Base  and  the  Rate  Per  Cent  given,  to  find 

the  Percentage. 

1.  How  much  is  5  %  of  800? 

Solutions. — 1.  Since  5  ^  =  Tfo,  5  %  of  800  =  of  800,  which 
is  40.  Or, 

2.  1  %  of  800  =  of  800,  which  is  8 ;  and  5  %  of  800  =  5  times 
8,  which  is  40. 

2.  What  is  6  %  of  1200  ?  8  %  of  250  ?  9  %  of  4000  ? 

3.  What  is  5  %  of  $300?  8  %  of  $450?  12  %  of  $500? 

4.  What  is  6  %  of  500  miles?  10%  of  250  miles?  15% 
of  600  miles  ?  20  %  of  300  miles  ? 


WRITTEN  PROBLEMS. 

5.  What  is  8  %  of  $674.50? 

1st  Process.  2d  Process. 

$674.50  $6,745  =  1  % 


.08 


8 


$53.96  00 


$53,960  =  8  % 


Note. — Let  the  pupil  use  one  method  until  he  is  familiar  with  it. 
C.Ar.— 12. 


138 


COMPLETE  ARITHMETIC. 


What  is 

6.  5%  of  245? 

16. 

4  %  of  $540  ? 

7. 

9%  of  360? 

17. 

*  %  of  $4000  ? 

8. 

15  %  of  1200  ? 

18. 

f  %  of  21700  ft.  ? 

9. 

25%  of  37.5? 

19. 

f%  of  $48.50? 

10. 

33%  of  $150? 

20. 

33 1  %  of  965  days? 

11. 

8%  of  $37.50? 

21. 

16f%  of  $.54?  ' 

12. 

3%  of  $180.25? 

22. 

15%  of  |?  Of  A? 

13. 

n  %  of  1050  lb.  ? 

23. 

66|%of*?  Of  2|? 

14. 

2|%  of  60.8  lb.? 

24. 

12%  of  .25?  Of  .45? 

15. 

12^%  of  560  days? 

25. 

6J%  of  50?  Of  .75? 

26.  What  is  the  difference  between  33  %  and  25J  %  of 
480  miles? 

27.  If  70  %  of  a  certain  ore  is  iron,  how  much  iron  is 
there  in  3740  pounds  of  ore  ? 

28.  If  20  %  of  air-dried  wood  is  water,  how  much  water 
is  there  in  143 J  tons  of  wood? 

29.  A  man  receives  $1650  a  year,  and  his  expenses  are 
87-J- %  of  his  income:  how  much  has  he  left? 

30.  A  grain  dealer  owning  58500  bushels  of  wheat, 
shipped  37^-  %  of  it  by  a  steamer,  33^  %  of  it  by  a 
schooner,  and  the  rest  of  it  by  railroad:  how  many  bushels 
did  he  ship  by  each? 

FORMULAS  AND  RULES. 

209.  Formulas. — 1.  Percentage —  base  X  rate  per  cent. 

2.  Amount  =  base  -f-  percentage. 

3.  Difference  —  base  — percentage. 

210.  Rules. — To  find  a  given  per  cent  of  any  number, 

1.  Multiply  the  given  number  by  the  rate  per  cent  expressed 
decimally.  Or, 

2.  Remove  the  decimal  point  two  places  to  the  left,  and  mul¬ 
tiply  the  result  by  the  rate. 

Note. — When  the  rate  is  an  aliquot  part  of  100,  the  per  cent  may 
be  found  by  taking  the  same  aliquot  part  of  the  base.  Thus,  331  tfc 
of  $48  =  1  of  $48.  The  process  in  each  of  the  succeeding  cases  may 
be  shortened  by  using  the  fraction  denoting  the  aliquot  part. 


PERCENTAGE. 


139 


Case  II. 

The  Base  and  the  Percentage  given,  to  find  the 

Rate  Per  Cent. 

1.  What  per  cent  of  16  is  . 4? 

Solutions. — 1.  1  is  A  of  16,  and  4  is  X4F,  or  .25  (Art.  121).  Hence, 
4  is  25  Jo  of  16.  Or, 

2.  1  oj0  of  16  is  .16,  and  4  is  as  many  per  cent  of  16  as  .16  is 
contained  times  in  4.00,  which  is  25. 

2.  What  per  cent  of  $50  are  $5?  $20? 

3.  What  per  cent  of  $300  are  $12?  $30? 

4.  What  per  cent  of  150  lb.  are  75  lb.?  50  lb.? 

5.  What  per  cent  of  250  ft.  are  50  ft.  ?  100  ft.  ? 


WRITTEN  PROBLEMS. 

6.  What  per  cent  of  62.5  is  15? 

\ 

1st  Process.  2d  Process. 

15  -4-  62.5  =  .24  1  cj0  of  62.5  =  .625 


.24  =  24^,,  Ans. 


15  -f-  .625  =  24,  Bate. 


Note. — The  quotient  obtained  by  the  first  process  is  the  rate  per 
cent,  and  the  quotient  obtained  by  the  second  process  is  the  rate. 


What  per  cent  of 

7.  75  is  4.5? 

8.  125  is  25? 

9.  120  is  40? 

10.  $450  are  $90? 

11.  $192  are  $32? 

12.  $760  are  $19? 

13.  $1000  are  $5? 

14.  $6  are  45  cts.  ? 


15.  75  lb.  are  16.5  lb.? 

16.  20  ft.  are  1.2  ft.  ? 

17.  37^  yd.  are  5  yd.  ? 

18.  .75  is  .15? 

19.  .60  is  .45? 

20. 

21. 

22. 


!  is 

4  is 


L9 

3  * 
19 


2*  is  f? 


23.  A  farmer  had  320  sheep  and  sold  48  of  them:  what 
per  cent  of  the  flock  did  he  sell? 

24.  A  gold  ring  is  22  carats  fine :  what  per  cent  of  it  is 
gold? 

25.  What  per  cent  of  $45  is  16|^  of  $150? 


140 


COMPLETE  ARITHMETIC. 


26.  A  regiment  of  750  men  lost  160  men  in  a  certain 
battle :  what  per  cent  of  the  regiment  remained  ? 

27.  What  per  cent  of  any  number  is  f  of  it?  f  of  it? 
5^  of  it?  ttq-  of  it? 

28.  What  per  cent  of  a  number  is  f  of  it?  2J  0 f  it? 
•J-f  of  it?  J  of  f  of  it? 

FORMULA  AND  RULES. 

211.  Formula. — Rate^o  —  'percentage  -f-  base. 

212.  Rules. — To  find  what  per  cent  one  number  is  of 
another, 

1.  Divide  the  number  which  is  the  percentage  by  the  base , 
and  the  quotient  expressed  in  hundredths  will  be  the  rate  per 
cent.  Or, 

2.  Divide  the  number  which  is  the  percentage  by  one  per 
cent  of  the  base ,  and  the  quotient  will  be  the  rate. 

Case  III. 

Percentage  and.  Rate  Per  Cent  given,  to  find  tHe 

Base. 

1.  45  is  15%  of  what  number? 

Solutions. — 1.  If  15%,  or  .15,  of  a  number  is  45,  the  number 
equals  45  h-  .15,  which  is  300.  Or, 

2.  If  45  is  15%  of  a  number,  1%  of  it  is  -fg-  of  45,  which  is  3,  and 
100%,  or  the  number,  is  100  times  3,  which  is  300. 

2.  320  is  16%  of  what  number? 

3.  7.2  pounds  are  12%  of  how  many  pounds? 

4.  Charles  is  15  years  old,  and  his  age  is  30%  of  his 
father’s  age :  how  old  is  his  father  ? 

5.  A  man’s  expenses  are  $28  a  month,  which  is  70%  of 
his  wages :  how  much  does  he  earn  a  month  ? 

6.  In  a  certain  school  56  pupils  study  arithmetic,  which 
is  28  per  cent  of  the  whole  number  of  pupils  in  school : 
how  many  pupils  in  the  school  ? 


PERCENTAGE. 


141 


WRITTEN  PROBLEMS. 

7.  A  man  owes  $4560,  which  is  30%  of  his  estate:  how 
much  is  his  estate? 

1st  Process.  2d  Process. 

$4560  -f-  .30  =  $15200,  Ans.  $4560  30  X  100  =  $15200,  Ans. 

8.  256  is  35%  of  what  number? 

9.  133^  is  16f  %  of  wThat  number? 

10.  107^-  is  15%  of  what  number? 

11.  540  sheep  are  36%  of  how  many  sheep? 

12.  5280  pounds  are  66J %  of  how  many  pounds? 

13.  $189.80  are  104%  of  what  sum  of  money? 

14.  $88.66  are  110%  of  what  sum  of  money? 

15.  The  number  of  pupils  in  daily  attendance  in  a  cer¬ 
tain  school  is  420,  which  is  80  %  of  the  number  enrolled : 
how  many  pupils  are  enrolled  ? 

16.  The  number  of  youth  of  school  age  in  a  certain  city 
is  5220,  which  is  36  %  of  the  number  of  inhabitants :  what 
is  the  population  of  the  city? 

17.  A  man  spent  60  %  of  his  money  for  a  suit  of  clothes, 
25  %  of  it  for  books,  and  had  $7.50  left:  how  much  money 
had  he  at  first? 

18.  A  man  invested  $5400  in  railroad  stock,  which  was 
37^%  of  his  property:  what  was  the  value  of  his  property? 

19.  In  a  storm,  a  ship’s  crew  threw  overboard  250  bar¬ 
rels  of  flour,  which  was  40  %  of  the  number  of  barrels  on 
board :  how  many  barrels  of  flour  were  left  on  board  ? 

20.  A  man  owning  60  %  of  a  factory,  sold  40  %  of  his 
share  for  $9600 :  at  this  rate,  what  was  the  value  of  the 
factory  ? 

21.  The  land  surface  of  the  earth  is  about  50000000 
sq.  miles,  which  is  33^  %  of  the  water  surface :  what  is 
the  extent  of  the  water  surface? 

22.  The  population  of  a  certain  city  in  1860  was  64000, 
which  is  80%  of  the  population  in  1870:  what  was  the 
population  in  1870? 


142 


COMPLETE  ARITHMETIC. 


FORMULA  AND  RULES. 

213.  Formula. — Base  =  'percentage  rate  per  cent 

214.  Rules. — To  find  a  number  when  a  certain  per  cent 
of  it  is  given. 

1.  Divide  the  number  which  is  the  percentage  by  the  rate  per 
cent  expressed  decimally.  Or, 

2.  Divide  the  number  which  is  the  percentage  by  the  rate ,  and 
multiply  the  quotient  by  100. 

Case  IV. 

The  Amount  or  Difference  and  the  Date  Per  Cent 
given,  to  find  the  Base. 

1.  216  is  8  %  more  than  what  number? 

Solutions. — 1.  If  216  is  8 %  more  than  a  number,  216  is  108%  or 
1.08  of  it,  and  hence  the  number  equals  216-4-1.08,  which  is  200.  Or, 

2.  If  216  is  108%  of  a  number,  1%  of  it  is  of  216,  which  is  2, 
and  100%  is  100  times  2,  which  is  200. 

2.  318  is  6  %  more  than  what  number? 

3.  $480  is  20  %  more  than  what  sum  of  money  ? 

4.  560  pounds  are  12%  more  than  how  many  pounds? 

5.  184  is  8  %  less  than  what  number  ? 

Suggestion. — If  184  is  8%  less  than  a  number,  184  is  100%  —  8%, 
or  92%  of  it. 

6.  285  is  5  %  less  than  what  number  ? 

7.  $356  are  11  %  less  than  how  many  dollars? 

8.  425  feet  are  15%  less  than  how  many  feet? 

9.  A  horse  cost  $160,  which  was  20  %  less  than  the  cost 
of  a  carriage :  wrhat  was  the  cost  of  the  carriage  ? 

10.  A  school  enrolls  230  boys,  which  is  15  %  more  than 
the  number  of  girls  enrolled :  how  many  pupils  in  the 
school  ? 

WRITTEN  PROBLEMS. 

11.  A  farm  was  sold  for  $6390,  which  was  12-J-  %  more 
than  it  cost :  what  was  the  cost  of  the  farm  ? 


PERCENTAGE. 


143 


1st  Process.  2d  Process. 

100^  +  12|%  =  1121#  =  1.125  $6390  112.5  =  $56.80  =  1  % 

$6390  -5- 1.125  =  $5680,  Ans .  $56.80  X 100  =  $5680,  Ans. 

12.  A  man’s  expenses  are  $400  a  year,  which  is  31J% 
less  than  his  income :  what  is  his  income  ? 

13.  276  is  15%  more  than  what  number? 

14.  What  number  increased  by  30  %  of  itself,  equals 
162.5? 

15.  What  number  diminished  by  16-f  %  of  itself,  equals 
2035.8? 

16.  A’s  farm  contains  306  acres,  which  is  32  %  less  than 
B’s :  how  many  acres  in  B’s  farm  ? 

17.  When  gold  was  worth  25  %  more  than  currency,  what 
was  the  gold  value  of  $150  in  currency? 

18.  When  gold  was  worth  50  %  more  than  currency, 
what  was  the  value  in  gold  of  a  dollar  bill  ? 

19.  The  number  of  pupils  in  daily  attendance  at  a  school 
is  570,  which  is  5  %  less  than  the  number  enrolled :  how 
many  pupils  are  enrolled? 

20.  The  number  of  pupils  enrolled  in  a  certain  town  is 
920,  which  is  15  %  more  than  the  average  number  of  pupils 
in  daily  attendance :  what  is  the  average  daily  attendance  ? 

21.  The  population  of  a  certain  city  in  1870  was  171572, 
w7hich  is  18  %  more  than  its  population  in  1860 :  what  was 
the  population  in  1860? 

FORMULAS  AND  RULES. 

215.  Formulas. — 1.  Base  —  amount  ~  (1  -f-  rate  %). 

2.  Base  =  difference  -f-  (1  —  rate  %). 

216.  Rules. — To  find  a  number  when  another  number 
is  given,  which  is  a  given  rate  per  cent,  more  or  less,  ' 

1.  Divide  the  given  number  by  1  plus  or  minus  the  given 
rate  per  cent  expressed  decimally.  Or, 

2.  Divide  the  given  number  by  100  plus  or  minus  the  given 
rate ,  and  multiply  the  quotient  by  100. 


144 


COMPLETE  ARITHMETIC. 


REVIEW  OF  THE  FOUR  CASES. 


217.  The  formulas  of  the  four  preceding  cases  of  per¬ 
centage  are  here  presented  together  for  comparison. 

Case  I. — Percentage  =  base  X  Tate  per  cent. 

Case  II.  — Rate  per  cent  =  percentage  -f-  base. 

Case  III. — Base  =  percentage  -f-  rate  per  cent. 

Case  IV. -Base  =  {  A™ffnt  +  £  +  mte  ?er 

(  Difference  -f-  (1  —  rate  per  cent). 


Note. — The  two  formulas,  amount  =  base percentage,  and  difference 
=  base — percentage ,  do  not  involve  the  operations  of  percentage,  but 
simply  the  adding  and  subtracting  of  numbers. 


MENTAL  PROBLEMS. 

1.  What  is  12 \<f0  of  640?  16f%  of  360? 

2.  What  is  33J%  of  672?  66f%  of  321? 

3.  15  is  what  per  cent  of  60  ?  Of  90  ? 

4.  16f  lb.  is  what  per  cent  of  50  lb.?  Of  100  lb.? 

5.  25%  of  120  is  what  per  cent  of  90? 

6.  33|-%  of  150  is  what  per  cent  of  250? 

7.  80  is  12 J%  of  what  number? 

8.  20%  of  105  is  25%  of  what  number? 

9.  33-|%  of  225  is  15%  of  what  number? 

10.  45  is  what  per  cent  of  75%  of  120? 

11.  360  is  20%  more  than  what  number? 

12.  60  is  33  J  %  more  than  what  number  ? 

13.  33 \%  240  is  33 J%  less  than  what  number? 

14.  25%  of  280  is  16f  %  more  than  what  number? 

15.  A  man  is  60  years  of  age,  and  20  %  of  his  age  is 
25  %  of  the  age  of  his  wife :  how  old  is  his  wife  ? 


WRITTEN  PROBLEMS. 

16.  The  population  of  a  certain  city  in  1860  was  63500, 
and  the  census  of  1870  shows  an  increase  of  17-J  % :  what 
was  the  population  in  1870? 


PERCENTAGE. 


145 


17.  A  farm  contains  480  acres,  of  which  30  %  is  meadow, 
25£  %  pasture,  16f  %  grain  land,  and  the  rest  woodland : 
how  many  acres  of  each  kind  of  land  in  the  farm? 

18.  A  merchant  failed  in  business  owing  $10500  and 
having  $6300  worth  of  property:  what  per  cent  of  his 
indebtedness  can  he  pay? 

19.  A  clerk,  receiving  a  yearly  salary  of  $950,  pays  $275 
a  year  for  board,  $180  for  clothing,  and  $150  for  other 
expenses:  what  per  cent  of  his  salary  is  left? 

20.  A  lady  pays  $280  a  year  for  board,  $175  a  year  for 
clothing  and  other  expenses,  and  lays  up  35  %  of  her  in¬ 
come:  what  is  her  income? 

21.  A  man’s  expenses  are  80%  of  his  income,  and  33^% 
of  his  income  equals  10  %  of  his  property,  which  is  valued 
at  $27000:  what  are  his  expenses? 

22.  A  merchant  sold  a  stock  of  goods  for  $10811,  and 
gained  13|- % :  what  was  the  cost  of  the  goods? 

23.  A  cargo  of  damaged  corn  was  sold  at  auction  for 
$9450,  which  was  33J  %  less  than  cost :  what  was  the  cost 
of  the  corn? 

24.  An  orchard  contains  1200  trees,  of  which  45  %  are 
apple,  22  %  peach,  12^  %  cherry,  and  the  rest  pear :  how 
many  trees  of  each  kind  in  the  orchard? 

25.  A  owns  42 \  %  of  a  factory  worth  $35000,  B  owns 
37  %  of  it,  and  C  owns  the  remainder :  what  is  the  value 
of  each  of  their  shares? 

26.  A  man  bequeathed  $7560  to  his  wife,  which  was 
62^  %  of  the  sum  bequeathed  to  his  children,  and  the  sum 
bequeathed  to  his  wife  and  children  was  80  %  of  his  estate : 
what  was  the  value  of  the  estate? 

27.  The  population  of  a  city  in  1870  was  41064,  which 
was  16%  more  than  in  1860,  and  the  population  in  1860 
was  6^  %  less  than  in  1865 :  what  was  the  population  in 
1865? 

28.  The  number  of  deaths  in  a  certain  city  in  1869  was 
1950,  which  was  equal  to  3J  %  of  the  population :  what  was 
the  population  ? 

C.Ar— 13. 


146 


COMPLETE  ARITHMETIC. 


APPLICATIONS  OF  PERCENTAGE. 

218.  The  principal  applications  of  percentage  are  Profit 
and  Loss,  Commission  and  Brokerage,  Capital  and  Stocks, 
Insurance,  Taxes,  Customs,  Bankruptcy ,  Interest,  Discount , 
Exchange,  Equation  of  Payments,  and  Equation  of  Accounts. 

All  the  problems  are  solved  by  the  application  of  one  or 
more  of  the  four  cases  of  percentage. 

PROFIT  AND  LOSS. 

219.  The  Cost  of  an  article  is  the  price  paid  for  it,  or 
the  total  expense  incurred  in  producing  it. 

220.  The  Selling  Price  of  an  article  is  the  amount 
asked  or  received  for  it  by  the  seller. 

The  selling  price  of  the  seller  is  the  cost  to  the  buyer,  and  vice  versa. 

221.  When  an  article  is  sold  for  more  than  its  cost,  it  is 
said  to  be  sold  at  a  profit  or  gain;  when  it  is  sold  for  less 
than  its  cost,  it  is  said  to  be  sold  at  a  loss  or  discount 
Hence, 

222.  Profit  or  Grain  is  the  amount  which  the  selling 
price  of  an  article  exceeds  its  cost. 

223.  Loss  or  J Discount  is  the  amount  which  the  sell¬ 
ing  price  of  an  article  is  less  than  its  cost. 

Note. — The  terms  gain  and  loss  are  not  limited  to  business  trans¬ 
actions.  When  any  quantity  undergoes  an  increase  or  decrease,  from 
any  cause,  there  is  a  gain  or  loss,  and  when  such  gain  or  loss  can  be 
expressed  in  hundredths,  it  may  be  computed  by  the  principles  of 
percentage. 


MENTAL  PROBLEMS. 

1.  A  merchant  bought  a  piece  of  cloth  for  $80,  and  sold 
it  at  25%  profit:  for  how  much  did  he  sell  it? 

2.  A  dealer  bought  hats  at  $5  apiece,  and  sold  them  at 
20  %  profit :  what  was  the  selling  price  ? 


PROFIT  AND  LOSS. 


147 


3.  Hats,  costing  $5  apiece,  were  sold  at  a  loss  of  20  °f0 : 
what  was  the  selling  price  ? 

4.  At  what  price  must  flour,  costing  $6  a  barrel,  be  sold 
to  gain  16  J  %  ? 

5.  A  grocer  bought  sugar  at  12  cts.,  16  cts.,  and  18  cts. 
a  pound:  for  how  much  must  each  kind  be  sold  to  gain 
20  %  ?  To  gain  25  %  ? 

6.  A  merchant  sells  broadcloth,  costing  84,  for  $5  a 
yard,  what  per  cent  does  he  gain  ? 

7.  When  broadcloth  costing  85  a  yard,  is  sold  for  84  a 
yard,  what  is  the  loss  per  cent  ? 

8.  Teas  costing  81.20  and  81.50  a  pound,  are  sold 
respectively  at  81.50  and  81.80  a  pound;  what  is  the  gain 
per  cent? 

9.  A  merchant  sold  velvet  at  a  profit  of  82  a  yard,  and 
gained  20  %  •  bow  much  did  it  cost  ? 

10.  A  dealer  sold  boots  at  81.50  a  pair  less  than  cost, 
and  thereby  lost  33^  %  :  what  did  they  cost  ? 

11.  A  grocer  sold  tea  at  30  cents  above  cost,  and  gained 
16 \cfo  •  what  was  the  cost  of  the  tea?  What  was  the  sell¬ 
ing  price? 

12.  A  man  sold  a  horse  for  890,  and  gained  20  %  •  what 
was  the  cost  of  the  horse  ? 

13.  A  man  sold  a  horse  for  880,  and  lost  20  %  :  what  was 
the  cost  of  the  horse? 

14.  Sold  butter  at  40  cts.  a  pound,  and  gained  25  %  • 
how  much  did  it  cost? 

15.  A  watch,  costing  880,  was  sold  at  a  loss  of  10  %  :  for 
how  much  wTas  it  sold  ? 

16.  How  must  shoes,  costing  82,  82.50,  and  83  a  pair,  be 
sold  respectively  to  gain  25  %  ?  How  must  each  kind  be 
sold  to  gain  30  %  ? 

17.  How  must  muslin  that  cost  10  cts.,  15  cts.,  and  18 
cts.  a  yard,  be  sold  to  gain  20  %  ? 

18.  Sold  tea  at  90  cts.  a  pound,  and  gained  20  %  :  what 
would  have  been  my  gain  per  cent  had  I  sold  it  at  81  a 
pound  ? 


148 


COMPLETE  ARITHMETIC. 


WRITTEN  PROBLEMS. 

19.  A  house  and  lot,  which  cost  $6750,  were  sold  at  a 
gain  of  12 f  %  :  for  how  much  were  they  sold  ? 

20.  Carriages,  costing  $165,  are  sold  at  18%  profit:  what 
is  the  gain  on  each  carriage? 

21.  A  man  paid  $4500  for  a  farm,  and  sold  it  for  $5400 : 
what  was  the  gain  per  cent? 

22.  A  drover  bought  cattle  at  $65  a  head,  and  sold  them 
at  $84.50  a  head:  what  was  the  gain  per  cent? 

23.  Carpeting,  costing  $1.75  a  yard,  is  sold  for  $2:  what 
is  the  gain  per  cent? 

24.  A  drover  bought  horses  at  $130  a  head,  expended  $6 
each  in  taking  them  to  market,  and  then  sold  them  at 
$153.50  a  head:  what  was  the  gain  per  cent? 

25.  A  cargo  of  wheat  costing  $16500,  being  damaged,  is 
sold  for  $13750:  what  was  the  loss  per  cent? 

26.  A  merchant  sold  a  lot  of  goods  at  12^-%  profit,  and 
gained  $8160:  what  was  the  cost? 

27.  A  grocer  sold  82  barrels  of  apples  at  22  %  profit,  and 
gained  $45.10:  what  was  the  cost  per  barrel? 

28.  A  man  sold  a  watch  for  $180,  and  lost  16f  %  :  what 
was  the  cost  of  the  watch? 

29.  A  house  and  lot  were  sold  for  $7762.50,  at  a  gain  of 
15%  :  what  was  the  cost? 

30.  A  dry  goods  firm  sold  $45000  worth  of  goods  in  a 
year ;  -J  of  the  receipts  were  sales  at  20  %  profit,  \  at  25  % 
profit,  and  the  rest  at  33^  %  profit :  what  was  the  cost  of 
all  the  goods? 

31.  Sold  a  piece  of  carpeting  for  $240,  and  lost  20  %  : 
what  selling  price  would  have  given  a  gain  of  20  %  ? 

32.  A  merchant  sells  goods  at  retail  at  30  %  above  cost, 
and  at  wholesale  at  12  %  less  than  the  retail  price :  what  is 
his  gain  per  cent  on  goods  sold  at  wholesale? 

33.  How  must  cloth,  costing  $3.50  a  yard,  be  marked 
that  a  merchant  may  deduct  15  %  from  the  marked  price 
and  still  make  15  %  profit  ? 


COMMISSION  AND  BROKERAGE. 


149 


34.  A  merchant  marked  a  piece  of  silk  at  25  %  above 
cost,  and  then  sold  it  at  20  %  less  than  the  marked  price : 
did  he  gain  or  lose,  and  how  much  ? 


FORMULAS  AND  RULES. 


224.  Formulas. — 1.  Gain  or  loss  =  cost  X  rate  %. 

2.  Hate  per  cent  =  gain  or  loss  ~  cost. 

3.  Cost  =  gain  or  loss  rate  %. 

( 1  -f-  rate  %  • 

\  1 — rate  %. 


4.  Cost  =  selling  price 


225.  Rules. — 1.  To  find  the  gain  or  loss  when  the  cost 
and  rate  per  cent  are  given,  Multiply  the  cost  by  the  rate  per 
cent  expressed  decimally.  (Form.  1.) 

2.  To  find  the  rate  per  cent  when  the  cost  and  the  gain 
or  loss  are  given,  Divide  the  gain  or  loss  by  the  cost ,  and  the 
quotient  expressed  in  hundredths  will  be  the  rate  per  cent 
(Form.  2.) 

3.  To  find  the  cost  when  the  gain  or  loss  and  the  rate 
per  cent  are  given,  Divide  the  gain  or  loss  by  the  rate  per 
cent  expressed  decimally.  (Form.  3.) 

4.  To  find  the  cost  when  the  selling  price  and  the  rate 
per  cent  of  gain  or  loss  are  given,  Divide  the  selling  price 
by  1  plus  or  minus  the  rate  per  cent.  (Form.  4.) 


Note. — Let  the  pupil  review  the  above  problems,  solving  those  in 
which  the  rate  is  an  aliquot  part  of  100,  by  using  the  fraction.  (Art. 
210,  Note.) 


COMMISSION  AND  BROKERAGE. 

226.  An  Agent  is  a  person  who  transacts  business  for 
another. 

227.  A  Factor  is  an  agent  who  buys  and  sells  goods 
intrusted  to  his  possession  and  control.  A  mercantile  factor 
is  also  called  a  Commission  Merchant. 

When  a  factor  lives  in  a  different  country  or  part  of  the  country 
from  his  employer,  he  is  called  a  Cori'espondent  or  Consignee.  The 


150 


COMPLETE  ARITHMETIC. 


goods  shipped  or  consigned  to  a  Consignee  are  called  a  Consignment ; 
and  the  sender  of  the  goods  is  called  a  Consignor. 

228.  A  Broker  is  a  person  who  buys  and  sells  gold, 
bills  of  exchange,  stocks,  bonds,  etc. ;  or  an  agent  who 
buys  and  sells  property  in  possession  of  others. 

229.  A  Collector  is  an  agent  who  collects  debts,  taxes, 
duties,  etc.  (Arts.  263,  267.) 

230.  Commission  is  an  allowance  made  to  a  factor  or 
other  agent,  for  the  transaction  of  business.  The  commis¬ 
sion  allowed  a  broker  is  called  Brokerage. 

231.  Commission  is  computed  at  a  certain  per  cent  of 
the  amount  of  property  bought  or  sold,  or  of  the  amount 
of  business  transacted.  The  rate  per  cent  is  called  the  Bate 
of  Commission ,  and  the  amount  of  business  transacted  is  the 
Base. 

The  rate  of  commission  varies  with  the  amount  and  nature  of  the 
business.  A  broker’s  commission  is  usually  less  than  a  factor’s. 

232.  The  'Net  Proceeds  of  a  sale  or  collection  are  the 
proceeds  less  the  commission  and  other  charges. 

MENTAL  PROBLEMS. 

1.  An  auctioneer  sold  $300  worth  of  furniture,  and 
charged  a  commission  of  5  %  :  how  much  did  he  receive  ? 

2.  A  peddler  bought  $500  worth  of  rags,  at  a  commis¬ 
sion  of  10%  :  what  was  his  commission? 

3.  An  agent  sold  $1200  worth  of  school  furniture,  at  a 
commission  of  16f%  :  how  much  did  he  receive? 

4.  An  attorney  collected  bad  debts  to  the  amount  of 
$800,  and  charged  20  %  commission :  what  was  his  com¬ 
mission  ? 

5.  A  society  paid  a  lad  $6  for  collecting  membership 
dues,  to  the  amount  of  $100:  what  rate  of  commission  did 
he  receive? 

6.  A  bookseller  received  $30  for  selling  $150  worth  of 
maps:  what  was  his  rate  of  commission? 


COMMISSION  AND  BROKERAGE. 


151 


7.  A  real-estate  broker  received  $40  for  selling  a  house 
and  lot,  at  5  %  commission:  for  how  much  was  the  prop¬ 
erty  sold? 

8.  An  attorney  received  $60  for  collecting  a  note,  at 
10%  commission:  what  was  the  amount  collected? 

9.  An  agent  received  $108,  with  which  to  buy  peaches, 
after  deducting  his  commission  at  8  % :  how  much  did  he 
expend  for  peaches? 

10.  A  factor  received  $309,  with  which  to  buy  flour, 
after  deducting  his  commission,  at  3  % :  what  was  the  cost 
of  the  flour? 

11.  A  lawyer  collected  a  bill  at  25  %  commission,  and 
remitted  $7.50  as  net  proceeds:  what  was  the  amount  col¬ 
lected?  What  was  the  lawyer’s  commission? 

12.  A  bookseller  sold  a  lot  of  books  on  commission,  at 
20%,  and  remitted  $160  as  net  proceeds:  for  how  much 
were  the  books  sold? 

WRITTEN  PROBLEMS. 

13.  A  commission  merchant  sold  540  barrels  of  flour,  at 
$6.37f  a  barrel:  what  was  his  commission  at  3  %  ? 

14.  A  real-estate  broker  sold  325  acres  of  land,  at  $24.50 
an  acre,  and  charged  a  commission  of  2f  %  :  what  was  his 
commission? 

15.  An  auctioneer  sold  $5160.50  worth  of  dry  goods,  and 
$715.25  worth  of  furniture:  what  was  his  commission,  at 

16.  A  lawyer  collected  65  %  of  a  note  of  $950,  and 
charged  6f  %  commission:  what  was  his  commission? 
What  was  the  amount  paid  over? 

17.  A  factor  in  New  Orleans  purchased  $75000  worth 
of  cotton  for  a  Lowell  manufacturer,  at  If  %  commission : 
what  was  his  bill  for  cotton  and  commission? 

18.  An  architect  charged  f%  for  plans  and  specifica¬ 
tions,  and  If  %  for  superintending  the  erection  of  a  build¬ 
ing,  costing  $120000:  how  much  was  his  fee? 


152 


COMPLETE  ARITHMETIC. 


19.  An  agent  furnished  a  school-house  for  $4500,  and 
received  $540  commission:  what  was  the  rate? 

20.  An  attorney  charged  $75  for  collecting  rents  to  the 
amount  of  $1125:  what  was  the  rate  of  commission? 

21.  A  commission  merchant  charged  2^  %  for  buying 
produce,  and  his  commission  was  $750 :  how  much  produce 
did  he  purchase? 

22.  A  wool  agent  received  5  %  for  buying  wool,  and  his 
commission  was  $208.50:  how  much  wool  did  he  buy? 

23.  My  agent  has  bought  3300  barrels  of  apples,  at 
$1.75  a  barrel,  and  I  allow  him  3%  commission:  how 
much  money  must  I  remit  to  pay  both  the  cost  of  the 
apples  and  the  commission? 

24.  A  Boston  merchant  sent  his  factor  in  Cincinnati 
$3529.20,  to  be  invested  in  bacon,  after  deducting  his  com¬ 
mission  at  2  %  :  how  much  did  he  expend  for  bacon,  and 
what  was  his  commission? 

25.  A  cotton  broker  in  Charleston  received  $11774,  with 
which  to  purchase  cotton,  after  deducting  his  commission  of 
1^  %  :  how  much  did  he  expend  for  cotton,  and  what  was 
his  commission? 

26.  A  merchant  paid  a  broker  -§  %  for  a  draft  of  $1280 
on  New  York:  how  much  was  the  brokerage? 

27.  A  broker  bought  $15600  worth  of  stocks,  and  charged 
\  %  :  what  was  his  fee  ? 

28.  A  real  estate  broker  sold  a  section  of  land  (640  A.) 
at  $7.50  an  acre,  and  invested  the  proceeds  in  railroad  stock, 
receiving  1^  %  for  selling  the  land  and  j  %  for  buying  the 
stock :  what  was  his  brokerage  ? 

29.  What  will  be  the  total  cost  of  750  yards  of  carpeting, 
at  $1.75  a  yard,  if  a  merchant  pays  2J  %  commission  for 
purchasing,  J  %  for  a  draft  covering  cost  and  agent’s  com¬ 
mission,  and  $12.50  for  freight? 

30.  A  grain  dealer  in  Chicago  received  $5000  with  direc¬ 
tions  to  purchase  wheat,  at  $1.10  a  bushel,  after  deducting 
his  commission  at  2J  %  :  how  many  bushels  of  wheat  did 
he  purchase? 


COMMISSION  AND  BROKERAGE. 


153 


31.  An  agent  sold  45  sewing  machines  at  $75  apiece, 
and  9  at  $125  apiece,  and,  deducting  his  commission,  re¬ 
mitted  $3375  to  the  manufacturer  as  proceeds:  what  was 
his  rate  of  commission  ? 

32.  A  factor  sold  $15000  worth  of  goods,  at  10  %  com¬ 
mission,  and  invested  the  proceeds  in  cotton,  first  deducting 
5  %  commission  for  buying :  how  much  money  did  he  invest 
in  cotton? 

33.  Smith  &  Jones  sell  for  C.  Bell  &  Co.  3040  pounds 
of  butter,  at  22  cts.  a  pound,  and  10560  pounds  of  cheese, 
at  15  cts.  a  pound,  and  invest  the  proceeds  in  dry  goods, 
first  deducting  their  commission  of  5  %  for  selling  and  3  % 
for  buying :  how  much  did  they  invest  in  dry  goods  ?  What 
wras  their  entire  commission? 

34.  A  commission  merchant  sold  1300  barrels  of  flour, 
at  $5.75  a  barrel,  receiving  a  commission  of  3^%,  and  in¬ 
vested  the  net  proceeds  in  coffee,  at  28  cts.  a  pound,  first 
deducting  2  %  commission :  how  many  pounds  of  coffee  did 
he  purchase?  What  was  his  entire  commission? 

FORMULAS  AND  RULES. 

233.  Formulas. — 1.  Com.  or  broJc.  =  base  X  Tate  %. 

2.  Rate  %  —  com.  or  brok.  -r-  base. 

3.  Base  =  com.  or  brok.  -f-  rate  % . 

4.  Base  =  ( base  -f  -  com.  or  brok.)  -f  -  (1  -f- 
rate  %.) 

234.  Rules. — 1.  To  find  commission  or  brokerage,  Mul¬ 
tiply  the  sum  of  money  denoting  the  amount  of  business  trans¬ 
acted ,  by  the  rate  per  cent  expressed  decimally.  (Form.  1.) 

2.  To  find  the  sum  to  be  invested  when  the  amount  given 
includes  both  the  sum  to  be  invested  and  the  commission 
or  brokerage,  Divide  the  given  amount  by  1  plus  the  rate  per 
cent  and  the  quotient  will  be  the  sum  to  be  invested.  (Form.  4.) 

Note.— These  two  rules  cover  all  the  ordinary  business  transac¬ 
tions  in  commission  or  brokerage,  but  the  pupil  should  be  required 
to  form  rules  embodying  each  of  the  four  formulas. 


154 


COMPLETE  ARITHMETIC 


CAPITAL  AND  STOCK. 

235.  Capital  is  property  invested  in  trade,  manufac¬ 
tures,  or  other  business. 

236.  The  Par  Value 

of  capital  is  its  original 
or  nominal  value. 

The  Market  Value 
of  capital  is  its  real  value, 
or  the  sum  for  which  it 
will  sell. 

When  the  market  value  oi 
capital  equals  its  par  value, 
it  is  said  to  be  at  'par ;  when 
the  market  value  is  more 
than  the  par  value,  it  is  above 
par ,  or  at  a  premium;  when 
the  market  value  is  less  than 
the  par  value,  it  is  below  par , 
or  at  a  discount. 

237.  Premium  is  the  amount  which  the  market  value 
of  capital  exceeds  its  par  value. 

238.  Discount  is  the  amount  which  the  market  value 
of  capital  is  less  than  its  par  value. 

239.  Premium  and  discount  are  computed  at  a  given  per 
cent  of  the  par  value.  The  rate  per  cent  is  called  the 
Rate  of  Premium,  or  the  Rate  of  Discount. 

240.  A  Company  is  an  association  of  persons  united 
for  the  transaction  of  business. 

The  association  of  several  persons  in  business  as  partners,  bound 
by  articles  of  agreement,  is  called  a  Partnership,  and  the  company  is 
commonly  called  a  Firm  or  House.  (Art.  380.) 

241.  An  Incorporated  Company  is  a  company 
organized  and  regulated  by  law.  It  is  called  a  Corporation, 
and  the  law  regulating  it  is  called  a  Charter. 


CAPITAL  AND  STOCK. 


155 


242.  The  capital  of  a  corporation  is  called  Stock,  and  is 
divided  into  equal  parts,  usually  $100  each,  called  Shares. 
The  owners  of  these  shares  are  called  Stockholders. 

243.  Certificates  of  StoeU  are  official  statements  of 
the  size  and  number  of  shares  owned  by  each  stockholder. 
They  are  called  Scrip,  and  are  bought  and  sold  like  other 
property. 

Stocks  are  at  par,  above  par,  or  below  par,  according  as  their  mar¬ 
ket  value  equals,  exceeds,  or  is  less  than  their  par  value  or  face. 

The  market  value  of  stocks  is  quoted  at  a  certain  per  cent,  of  the 
par  value.  Stocks  quoted  at  108  are  worth  108  %  of  their  face,  that 
is,  are  8%  above  par ;  stocks  quoted  at  92  are  worth  92%  of  their  face, 
that  is,  are  8%  below  par. 

The  business  of  buying  and  selling  stocks  is  called  Stock  Jobbing, 
and  persons  engaged  in  such  business  are  called  Stock  Jobbers,  or  Stock 
Brokers. 

244.  The  Gross  Earnings  of  a  company  are  the 
total  receipts  from  its  business;  and  the  Net  Earnings  are 
the  net  profits,  found  by  deducting  all  expenses  and  losses 
from  the  gross  earnings. 

245.  A  Dividend  is  the  part  of  the  earnings  of  a 
company  distributed  among  the  stockholders. 

Dividends  are  usually  declared  annually  or  semi-annually,  and 
they  are  computed  as  a  per  cent  of  the  par  value  of  the  stock.  The 
rate  per  cent  is  called  the  Bate  of  Dividend. 

246.  An  Assessment  is  a  sum  levied  upon  the  stock¬ 
holders  to  meet  the  losses  or  expenses  of  the  business. 

The  business  of  incorporated  companies  is  usually  managed  by 
directors,  who  are  elected  by  the  stockholders,  each  being  entitled  to 
as  many  votes  as  he  owns  shares. 

Note. — When*  a  business  corporation  wishes  to  raise  money  in  ad¬ 
dition  to  that  derived  from  its  capital  stock,  it  issues  notes  or  bonds, 
payable  at  a  specified  time  with  interest,  and  secured  by  mortgage  on 
the  property  of  the  corporation.  These  notes  are  called  Mortgage 
Bonds,  and  their  owners  are  called  Bondholders.  These  bonds  are 
negotiable  and  are  called  Stocks  (Art.  326),  but  they  should  be  care¬ 
fully  distinguished  from  Capital  Stock. 


156 


COMPLETE  ARITHMETIC. 


MENTAL  PROBLEMS. 

1.  When  stock  is  6  %  premium,  what  is  the  market 
value  of  $1  ?  Of  $100  ? 

2.  When  stock  is  12  %  discount,  what  is  the  market 
value  of  $1?  Of  $100? 

S.  How  much  will  5  shares  of  telegraph  stock  cost,  at 

4  %  premium  ?  At  4  %  discount  ? 

Note. — A  share  is  $100,  if  no  other  value  is  named. 

4.  How  is  stock  quoted  when  it  is  15  %  premium  ? 
When  it  is  15  %  discount  ? 

5.  When  stock  is  quoted  at  107J,  what  is  the  value  of 
$1?  Of  $100? 

6.  When  stock  is  quoted  at  87,  what  is  the  value  of  $1? 
Of  $100? 

7.  How  much  will  10  shares  of  mining  stock  cost  when 
quoted  at  104?  At  85? 

8.  When  bank  stock  is  quoted  at  105,  how  many  shares 
can  be  bought  for  $525?  For  $840? 

9.  A  company  declares  a  dividend  of  3  %  :  how  much 
will  a  stockholder,  owning  15  shares,  receive? 

10.  A  manufacturing  company  made  an  assessment  of 

5  % ,  to  repair  damages  caused  by  a  freshet :  how  much 
must  a  stockholder,  owning  20  shares,  pay? 

WRITTEN  PROBLEMS. 

11.  A  man  bought  75  shares  of  railroad  stock  at  7^  % 
discount:  how  much  did  they  cost? 

12.  Bought  100  shares  of  Little  Miami  stock  at  109^, 
and  sold  them  at  112^:  how  much  did  I  gain  in  the  trans¬ 
action  ? 

13.  A  broker  bought  70  shares  of  insurance  stock  at 
6J-  %  premium,  and  sold  them  at  f  %  discount :  how  much 
did  he  lose? 


CAPITAL  AND  STOCK. 


157 


14.  A  man  bought  52  shares  of  Illinois  Central  at  127 ; 
and  sold  36  shares  at  135,  and  the  rest  at  137£:  how  much 
did  he  gain  ? 

15.  A  man  exchanged  52  shares  of  railroad  stock  at  80, 
for  insurance  stock  at  104 :  how  many  shares  of  insurance 
stock  did  he  receive? 

16.  A  broker  bought  84  shares,  $50  each,  of  telegraph 
stock  at  94,  and  sold  them  at  lOOf:  how  much  did  he 
gain  ? 

17.  The  Cincinnati  Gas  Co.  declares  a  dividend  of 
16f  %  :  how  much  will  a  man  holding  36  shares  receive  ? 

18. ^  The  capital  of  an  insurance  company  is  $500000, 
and  it  declares  a  dividend  of  4|%  :  how  much  money  is 
distributed  among  the  stockholders? 

19.  A  company  with  a  capital  of  $125000  declares  a  div¬ 
idend  of  4  % ,  with  $3500  surplus :  what  were  the  net  earn¬ 
ings  of  the  company? 

Note. — The  surplus  is  a  part  of  the  net  earnings  set  apart  to  meet 
future  demands. 

20.  The  entire  capital  stock  of  the  railroads  in  Ohio  for 
1869  was  $106686116,  and  their  net  earnings  for  the  year 
were  $9051998:  what  was  the  average  rate  of  dividend? 

21.  The  net  earnings  of  a  gas  company  are  $22425,  and 
the  capital  stock  is  $215000:  what  rate  of  dividend  can  be 
declared,  no  surplus  being  reserved  ?  What  will  be  the  div¬ 
idend  on  45  shares? 

22.  The  capital  of  a  mining  company  is  $450000;  the 
gross  receipts  are  $70680 ;  and  the  expenses  are  $40325 : 
what  rate  of  dividend  can  it  declare,  reserving  a  surplus 
of  $6505  ? 

23.  How  many  shares  of  bank  stock  at  4  %  premium, 
can  be  bought  for  $8320? 

24.  How  much  railroad  stock,  at  12^-  discount,  can  be 
bought  for  $8750? 

25.  When  N.  Y.  Central  is  quoted  at  95-J,  how  much 
stock  can  be  bought  for  $6894,  brokerage  -§-  %  ? 


158 


COMPLETE  ARITHMETIC. 


26.  A  broker  bought  84  shares  of  coal  stock,  at  108J,  re¬ 
ceived  a  dividend  of  5%%,  and  then  sold  the  stock  for  106: 
how  much  did  he  gain? 

27.  A  broker  bought  stock  at  4  %  discount,  and,  selling 
the  same  at  5  %  premium,  gained  $450:  how  many  . shares 
did  he  purchase  ? 

28.  A  man  bought  Michigan  Central  at  120,  and  sold  at 
124 :  what  per  cent  of  the  investment  did  he  gain  ? 

FORMULAS  AND  RULES. 

247.  F ormul  as. — 1 .  Dividend  or  assessment = stock  X  rate  %  • 

2.  Rate  %  =  divid.  or  assess,  -f-  stock. 

3.  Stock  =  divid.  or  assess.  —  rate  %. 

4.  Prem.  or  dis.  =par  value  X  rate%. 

5.  Rate  %  —  prem.  or  dis.  -s-  par  value. 

6.  Par  value— prem.  or  dis.  ~  rate  %. 

7.  Par  val.  =  market  val.  —  (1  ±  rate  %  ). 

8.  Market  val.  =  par  val.  X  (1  ±  rate%). 

9.  Market  val.  —par  val. -{-prem.  or  —  dis. 

Note. — Formulas  4,  7,  and  8  cover  all  ordinary  transactions  in 
stock  jobbing.  The  sign  dr,  used  in  7  and  8,  is  read  plus  or  minus. 

248.  Rules. — 1.  To  find  the  dividend  or  assessment  when 
the  stock  and  rate  per  cent  are  given,  Multiply  the  stock  by 
the  rate  per  cent  expressed  decimally.  (Form.  1.) 

2.  To  find  the  rate  per  cent  when  the  dividend  or  assess¬ 
ment  and  total  stock  are  given,  Divide  the  dividend  or  assess¬ 
ment  by  the  amount  of  stock,  and  the  quotient,  expressed  in 
hundredths ,  will  be  the  rate  per  cent.  (Form.  2.) 

3.  To  find  the  stock  when  the  dividend  or  assessment  and  ' 
the  rate  per  cent  are  given,  Divide  the  dividend  or  assessment 
by  the  rate  per  cent  expressed  decimally.  (Form.  3.) 

4.  To  find  the  premium  or  discount  on  a  given  amount 
of  stock,  Multiply  the  amount  of  stock  by  the  rate  per  cent  ex¬ 
pressed  decimally.  (Form.  4.) 


INSURANCE. 


159 


5.  To  find  the  cost  or  market  value  of  a  given  amount 
of  stock,  Multiply  the  amount  of  stock  (1)  by  1  plus  or  minus 
the  rate  per  cent  (Form.  8)  ;  or  (2)  by  the  quoted  price  ex¬ 
pressed  as  hundredths. 

Note. — The  cost  may  also  be  found  by  adding  the  premium  to,  or 
subtracting  the  discount  from,  the  par  value.  (Form.  9.) 

6.  To  find  the  amount  of  stock  which  can  be  bought  for 
a  given  amount  of  money,  Divide  the  amount  of  money  to  be 
invested  (1)  by  1  plus  or  minus  the  rate  per  cent  (Form.  7); 
or  (2)  by  the  quoted  price  expressed  as  hundredths. 

Note. — When  brokerage  is  paid,  the  rate  of  brokerage  must  be 
added  to  the  quoted  price  before  dividing. 


INSURANCE. 


249.  Insurance  is  a  guaranteed  indemnity  for  loss. 

250.  Fire  Insur¬ 
ance  is  a  guaranteed  in¬ 
demnity  for  loss  of  prop¬ 
erty  by  fire. 

251.  Marine  In¬ 
surance  is  a  guaran¬ 
teed  indemnity  for  loss 
of  property  while  trans¬ 
ported  by  water. 


Insurance  on  the  property 
transported  is  called  Cargo  In¬ 
surance;  that  on  the  vessel  is 
called  Hull  Insurance. 

252.  life  Insurance  is  a  guaranteed  indemnity  for 
loss  of  life. 

Health  Insurance  guarantees  the  insured  a  certain  sum  of  money 
if  sick ;  and  Accident  Insurance  pledges  a  like  indemnity  if  the  in¬ 
sured  is  injured  by  accident. 


160 


COMPLETE  ARITHMETIC. 


253.  The  Policy  is  the  written  contract  between  the 
insurer  and  the  insured. 

The  insurer  is  called  an  Underwriter ,  and  the  insured  a  Policy 
Holder . 

254.  The  Premium  is  the  sum  paid  by  the  insured 
to  obtain  the  insurance.  It  is  a  specified  per  cent  of  the 
amount  insured. 

The  act  of  insuring  is  called  taking  a  risk.  When  property  is  in¬ 
sured,  the  valuation  or  amount  is  usually  made  less  than  the  real 
value  of  the  property. 

The  insurance  business  is  chiefly  carried  on  by  corporations,  called 
Insurance  Companies.  In  Joint  Stock  Companies  the  profits  and  losses 
are  shared  by  the  stockholders,  but  in  Mutual  Companies  they  are 
divided  among  the  policy  holders. 


MENTAL  PROBLEMS. 

1.  A  house  was  insured  for  $2500,  at  1  %  :  what  was 
the  premium  ? 

2.  A  stock  of  goods  was  insured  for  $8000,  at  J  %  :  what 
was  the  premium  ? 

3.  A  hotel  worth  $6000  is  insured  for  -§  of  its  value,  at 
li  %  •  what  is  the  premium  ? 

4.  What  will  it  cost  to  get  a  house  insured  for  $4000, 
for  10  years,  at  J  %  a  year  ? 

5.  The  premium  paid  for  insuring  a  library  for  $500,  is 
$5 :  what  is  the  rate  of  insurance  ? 

6.  An  insurance  company  insures  a  school  house  for 
$10000,  and  charges  $50  premium:  what  is  the  rate? 

7.  The  premium  for  insuring  a  cargo  of  goods,  at  2%, 
was  $240  :  what  was  the  amount  of  goods  insured  ? 

WRITTEN  PROBLEMS. 

8.  A  factory  worth  $75000  is  insured  for  -J  of  its  value, 
at  1^  °f0  :  how  much  is  the  premium  ? 

9.  A  merchant  has  his  store  insured  for  $7850,  at  f 
and  his  goods  for  $12400,  at  \  %  :  what  premium  does  he  pay? 


INSURANCE. 


161 


10.  A  house  worth  $5400  was  insured  for  §  of  its  value, 
at  and  the  cost  of  the  survey  and  policy  was  $1.50: 
what  was  the  cost  of  the  insurance? 

Note. — When  a  new  risk  is  taken,  a  small  fee  is  usually  charged 
for  examining  the  property,  called  the  Survey ,  and  for  issuing  the 
policy. 

11.  The  owners  of  a  vessel  paid  $561  for  a  hull  insurance 
of  $25500 :  what  was  the  rate  of  insurance  ? 

12.  A  merchant  paid  $100  for  an  insurance  of  $12500 
on  a  stock  of  goods:  what  was  the  rate  of  insurance? 

13.  A  school  house  is  insured  at  §  %,  and  the  premium 
was  $93.60:  for  how  much  is  the  house  insured? 

14.  A  grain  shipper  paid  $525  for  the  insurance  of  a 
cargo  of  wheat,  at  1^  %  :  for  how  much  was  the  wheat  in¬ 
sured  ? 

15.  A  company  insured  a  block  of  buildings  for  $150000, 

at  |  % ,  but,  the  risk  being  too  great,  it  re-insured  $40000  in 
another  company,  at  -J  %,  and  $35000  in  another  company, 
at  4  %.  How  much  premium  did  it  receive  more  than  it 
paid  ?  • 

16.  A  block  of  buildings  worth  $135000  is  insured  for  4 
of  its  value  by  three  companies,  the  first  taking  ^  of  the 
risk  at  J  </0  ;  the  second  taking  f  of  it  at  J  %  ;  and  the 
third  taking  the  remainder  at  J  % .  What  was  the  total 
premium  ? 

17.  Suppose  the  above  block  should  be  damaged  by  fire 
to  the  amount  of  $60000,  how  much  of  the  damage  would 
each  company  be  obliged  to  pay  ? 

18.  A  house  which  has  been  insured  for  $3500  for  10 
years,  at  4  °f0  a  year,  was  destroyed  by  fire :  how  much  did 
the  money  received  from  the  company  exceed  the  cost  of 
premiums  ? 

19.  A  steamer,  burned  in  1869,  had  been  insured  by  a 
single  company  20  years,  for  $40000,  at  24  %  a  year :  what 
was  the  actual  loss  to  the  company,  no  allowance  being  made 
for  interest  ? 

20.  A  grain  dealer  had  a  cargo  of  wheat,  valued  at 

C.Ar.— 14. 


162 


COMPLETE  ARITHMETIC. 


$31360,  insured  at  2  %,  so  as  to  cover  both  the  value  of 
the  wheat  and  the  cost  of  the  premium :  for  how  much  was 
the  wheat  insured  ? 

Suggestion. — Since  the  premium  was  2  %  of  the  amount  insured, 
the  value  of  the  wheat  was  100  fo — 2%,  or  98  fo  of  the  amount  in¬ 
sured.  Hence,  $31360  =  j-fo  of  the  amount  insured.  (Case  IV.) 

21.  For  how  much  must  a  cargo  of  lumber,  worth  $21825, 
be  insured,  at  3  % ,  to  cover  both  the  value  of  the  lumber  and 
the  cost  of  the  premium  ? 

22.  For  how  much  must  property,  worth  $11859.40,  be 
insured,  at  1^  % ,  to  cover  both  property  and  premium  ? 

23.  For  what  must  a  cargo  of  goods,  valued  at  $11520, 
be  insured,  at  4  %,  to  cover  both  goods  and  premium? 

24.  What  amount  must  be  insured  to  cover  property 
worth  $2587,  and  premium  at  \  %  ? 

25.  To  cover  both  goods  and  premium  at  1%,  a  mer¬ 
chant  had  a  cargo  of  goods  insured  for  $35000 :  what  was 
the  value  of  the  goods? 

26.  A  merchant  shipped  a  cargo  of  flour  from  New  York 
to  Liverpool,  and,  to  cover  both  the  flour  and  the  premium, 
he  took  out  a  policy  for  $50400,  at  \  what  was  the 
value  of  the  flour  ? 

FORMULAS  AND  RULES. 

255.  F ormulas. — 1 .  Premium  =  amount  insured  X  rate  % . 

2.  Rate  %  =prem.  -f-  amount  insured. 

3.  Amount  insured  —  prem.  -r-  rate  % . 

4.  Property  and  premium  =  property  — 
(1  —  rate  %). 

256.  Rules. — 1.  To  find  the  premium,  Multiply  the  amount 
insured  by  the  rate  per  cent  expressed  decimally.  (Form.  1.) 

2.  To  find  for  what  sum  property  must  be  insured  to 
cover  both  property  and  premium,  Divide  the  value  of  the 
property  insured  by  1  less  the  rate  per  cent  expressed  deci¬ 
mally.  (Form.  4.) 


LIFE  INSURANCE. 


163 


LIFE  INSURANCE. 

257.  In  Life  Insurance  the  insurer  agrees  to  pay 
to  the  heirs  of  the  insured,  or  to  some  person  named  in  the 
policy,  a  stipulated  sum  on  the  death  of  the  insured,  or  at 
a  specified  time  should  his  death  not  occur  before. 

258.  When  the  policy  matures  at  the  death  of  the  in¬ 
sured,  it  is  called  a  Life  Policy;  when  it  matures  in  a  speci¬ 
fied  number  of  years,  it  is  called  a  Term  Policy  or  an  Endow¬ 
ment  Policy. 

The  premium  in  life  insurance  may  be  paid  in  a  single  payment  ; 
or  it  may  be  paid  annually  during  the  term  of  the  policy ;  or  it 
may  be  paid  annually  for  a  specified  number  of  years,  usually  for 
ten  years. 

A  Non-forfeiting  Policy  guarantees  the  insured  an  equitable  part  of 
the  sum  insured  in  case  he  should  fail  to  pay  his  annual  premiums 
after  a  specified  number  of  payments  have  been  made. 

259.  The  premium  is  computed  at  a  certain  sum  or  rate 
per  $1000  insured,  the  rate  varying  with  the  age  of  the  in¬ 
sured  at  the  time  the  policy  is  issued. 

The  basis  on  which  the  rate  of  life  insurance  is  determined  is  the 
expectation  of  life ,  or  the  average  extension  of  life  beyond  the  given 
age,  as  shown  by  life  statistics.  Tables  have  been  formed  showing 
the  expectation  of  life  for  every  year  of  man’s  age.  (See  appendix.) 


WRITTEN  PROBLEMS. 

27.  A  man  45  years  of  age  has  his  life  insured  for  $3000, 
at  $37.30  per  $1000:  what  annual  premium  does  he  pay? 

28.  A  man  30  years  of  age  has  his  life  insured  for  $6000, 
at  $23.60  per  $1000:  what  is  his  annual  premium? 

29.  A  man  38  years  of  age  is  insured  for  $5000  on  the  ten 
year  plan,  at  $44.50  per  $1000:  what  will  be  the  sum  of  his 
premiums  should  they  all  be  paid  ? 

30.  A  man  35  years  of  age  took  out  a  life  policy  for 
$4000,  at  the  rate  of  $27.50  per  $1000 ;  he  died  at  the  age 


164 


COMPLETE  ARITHMETIC. 


of  60 :  how  much  greater  was  the  amount  insured  than  the 
sum  of  the  annual  payments? 

31.  A  man  27  years  of  age  took  out  a  life  policy  for 
$8000,  for  the  benefit  of  his  wife,  at  the  rate  of  $21.70 
per  $1000,  and  his  death  occurred  at  the  age  of  33:  how 
much  did  the  widow  receive  more  than  had  been  paid  in 
annual  premiums? 


TAXES. 

260.  A  Tax  is  a  sum  of  money  assessed  on  the  person 
of  a  citizen,  or  on  property,  or  business  for  the  support  of 
government  or  other  public  purposes. 

261.  A  tax  on  the  person  of  a  citizen  is  called  a  Poll 
Tax ,  or  Capitation  Tax. 

232.  A  tax  on  property  is  called  a  Property  Tax.  It  is 
assessed  either  at  a  given  rate  per  cent  of  the  valuation,  or 
at  the  rate  of  a  given  number  of  mills  on  the  dollar. 

Property  is  classified  as  Real  Estate  and  Personal  Property ,  the 
former  including . all  fixed  property,  as  houses  and  lands;  and  the 
latter,  all  movable  property.  The  taxable  value  of  real  estate  is 
appraised  by  officers  called  Appraisers  or  Assessors,  and  the  value 
of  personal  property  is  fixed  by  the  owner,  under  oath,  or  by  the 
assessor. 

263.  A  tax  on  the  annual  income  of  a  citizen  or  corpora¬ 
tion  is  an  Income  Tax;  a  tax  on  business,  an  Excise  Tax; 
and  a  tax  on  imported  goods,  a  Custom  or  Duty.  (Art.  267.) 

Income  taxes  are  assessed  at  a  given  rate  per  cent  of  annual 
net  incomes,  less  specified  exemptions  and  deductions.  Excise  taxes 
consist  of  fees  for  business  licenses,  revenue  stamps  for  business 
papers,  taxes  on  manufactured  products,  etc. 

The  Internal  Revenue  of  the  United  States  is  chiefly  derived  from 
excise  taxes,  assessed  and  collected  by  United  States  officers,  called 
Assessors  and  Collectors.  No  income  taxes  have  been  collected  in 
this  country  since  1872. 

Note. — Taxes  are  classified  as  direct  and  indirect.  Property  and 
poll  taxes  are  direct;  and  excise  taxes  and  duties  are  indirect,  since 
they  are  paid  indirectly  by  the  consumer. 


TAXES. 


165 


WRITTEN  PROBLEMS. 


1.  The  valuation  of  the  taxable  property  of  a  village  was 
8632000,  and  a  tax  of  89480  was  assessed  to  build  a  school 
house :  what  was  the  rate  of  tax  ? 


Process.  Since  the  tax  was  .015  of  the  property, 

$9480  -i-  $632000  =  .015  the  rate  was  1.5%,  or  15  mills  on  the 
Rate  =  1^%,  or  15  mills.  dollar. 

2.  The  tax  levied  in  a  certain  city,  for  all  purposes,  was 
$259776,  and  the  taxable  property  was  listed  at  $21648000: 
what  was  the  rate  of  tax  in  mills? 

3.  The  amount  of  tax  to  be  assessed  in  a  certain  township 
is  $19340.20;  the  taxable  property  is  $1425400;  and  the 
number  of  polls,  assessed  at  $1.50  each,  is  540:  what  rate 
of  tax  must  be  assessed  on  property? 

4.  The  cost  of  the  public  schools  of  a  certain  city  for  the 
next  school  year,  is  estimated  at  $36848 :  what  amount  of 
school  tax  must  be  assessed,  the  cost  of  collecting  being 
2  % ,  and  allowing  6  %  of  the  assessed  tax  to  be  uncol¬ 
lectible  ? 


Process. 

.98)  $36848 

.94  )  $37600,  Tax  collected. 
$40000,  Tax  assessed. 


Since  2%  of  tax  collected  is  paid  for 
collection,  $36848  is  98%  or  .98  of  the 
tax  to  be  collected.  $36848  is  .98  of 
$37600.  (Case  IV.)  Since  6  %  of  the 
tax  assessed  is  not  collectible,  the  col¬ 


lectible  tax,  or  $37600,  is  .94  of  the  tax  to  be  assessed.  $37600  is  .94 
of  $40000.  Hence,  $40000  is  the  amount  to  be  assessed. 


Note. — Since  the  amount  of  uncollectible  tax  can  only  be  esti¬ 
mated,  the  amount  to  be  assessed  may  be  found,  with  sufficient  accu¬ 
racy  for  all  practical  purposes,  by  adding  the  percentages  for  collec¬ 
tion  and  for  uncollectible  taxes  to  the  amount  of  money  to  be  raised 
for  the  given  purpose. 

5.  The  net  proceeds  of  a  certain  tax  assessment,  after 
deducting  \\%  f°r  collection,  was  $11703.84£,  and  7 

of  the  tax  was  not  collected:  what  was  the  amount  of  tax 
assessed  ? 

6.  The  amount  of  tax  assessed  on  the  property  of  a  cer- 


166 


COMPLETE  ARITHMETIC. 


tain  city  was  $145850;  the  treasurer  was  allowed  a  fee  of 
f  %  f°r  collection,  and  10%  of  the  tax  was  uncollectible: 
what  were  the  net  proceeds  of  the  assessment? 

7.  The  taxable  property  of  a  certain  city  is  valued  at 
$87045060,  and  the  rate  of  tax  for  school  purposes  is  5^ 
mills  on  the  dollar:  what  is  the  amount  of  school  tax  as¬ 
sessed? 

Suggestion. — Since  5|  mills  =  .005^  of  a  dollar,  the  tax  assessed 
=  .005^  of  the  property. 

8.  The  valuation  of  taxable  property  in  a  certain  county 
is  $35460850,  and  the  rate  of  tax  levied  is  25  mills:  what 
will  be  the  net  proceeds  of  the  tax,  the  cost  of  collection 
being  3%,  and  8  %  of  the  tax  being  uncollectible? 

9.  When  the  rate  of  taxation  is  15  mills,  what  is  the 
amount  of  tax  on  A’s  property,  listed  at  $13560?  On  B’s, 
listed  at  $9850.60?  On  C’s,  listed  at  $50060? 

10.  A  man’s  net  income  is  $2750,  of  which  $1354  is  by 
law  exempt  from  taxation:  what  is  his  income  tax  at  5%  ? 
At  3%? 

11.  A  man’s  income  is  $3570,  and  the  deductions  allowed 
by  law  amount  to  $1650:  what  is  his  income  tax  at  5%  ? 

12.  A  man  pays  a  tax  of  12 \  mills  on  his  property, 
listed  at  $9850,  and  an  income  tax  of  5%  on  a  net  income 
of  $2750:  what  is  his  total  annual  tax? 

FORMULAS  AND  RULES. 

264.  Formulas. — 1.  Tax  —  property  X  rate  % . 

2.  Rate  %=tax  -f-  'property. 

3.  Tax  collected  =  net  proceeds  -s-  (1  — 
rate  %  for  collection). 

4.  Tax  assessed  =  tax  collected  -j-  (1  — 
rate  %  of  tax  uncollected). 

265.  Rules. — 1.  To  find  the  amount  of  tax,  Multiply  the 
amount  of  taxable  property  by  the  rate  of  tax,  expressed  deci¬ 
mally. 


TAXES. 


167 


2.  To  find  the  rate  of  tax  in  mills,  Divide  the  amount  of 
tax  by  the  amount  of  property,  and  express  the  quotient  as  thou¬ 
sandths.  The  number  of  thousandths  will  be  the  number  of  mills. 

TAX  TABLES. 

266.  The  labor  of  making  out  a  tax  list  may  be  much 
lessened  by  using  tables  giving  the  tax  on  convenient 
amounts  of  property,  at  the  given  rate. 


Table  for  a  Rate  of  15  mills. 


PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

$1 

$.015 

$10 

$0.15 

$100 

$1.50 

$1000 

$15. 

$10000 

$150. 

2 

.03 

20 

.30 

200 

3.00 

2000 

30. 

20000 

300. 

3 

.045 

30 

.45 

300 

4.50 

3000 

45. 

30000 

450. 

4 

.06 

40 

.60 

400 

6.00 

4000 

60. 

40000 

600. 

5 

.075 

50 

.75 

500 

7.50 

5000 

75. 

50000 

750. 

6 

.09 

60 

.90 

600 

9.00 

6000 

90. 

60000 

900. 

7 

.105 

70 

1.05 

700 

10.50 

7000 

105. 

70000 

1050. 

8 

.12 

80 

1.20 

800 

12.00 

8000 

120. 

80000 

1200. 

9 

.135 

90 

1.35 

900 

13.50 

9000 

135. 

90000 

1350. 

13.  Find  by  the  above  table  the  tax  on  $875.64,  at  the 
rate  of  15  mills. 


Process. 


$875.64  =  $800  +  $70  +  $5  +  $.60  +  $.04. 


Tax  on  - 

'$800.  =$12.00 

70.  =  1.05 

5.  =  .075 

.60  =  .009 

.04  =  .0006 

Tax  on 

$875.64  =  $13.1346 

Since  $60  are  100  times 
60  cts.,  the  tax  on  60  cts. 
is  found  by  dividing  the 
tax  on  $60  by  100,  which 
is  done  by  removing  the 
decimal  point  two  places 
to  the  left.  The  tax  on  4 
cts.  is  found,  in  like  man¬ 
ner,  from  the  tax  on  $4. 


Find  by  the  above  table  the  tax  of 


14.  Mr.  A  on  $708. 

15.  Mr.  B  on  $960. 

16.  Mr.  C  on  $85.80. 

17.  Mr.  D  on  $3405. 

18.  Mr.  E  on  $860.50. 


19.  Mr.  F  on  $5408. 

20.  Mr.  G  on  $85600. 

21.  Mr.  IT  on  $90908. 

22.  Mr.  I  on  $150340. 

23.  Mr.  J  on  $225350. 


168 


COMPLETE  ARITHMETIC. 


CUSTOMS  OR  DUTIES. 


267.  Customs  are  taxes  levied  by  the  national  govern¬ 


ment  on  imported  goods 
and  the  tonnage  of  ves¬ 
sels.  Customs  are  also 
called  Duties. 

Ports  of  Entry  for  foreign 
goods  are  established  by  law, 
and  at  each  port  of  entry  there 
is  a  Custom  House ,  where  cus¬ 
toms  or  duties  are  collected. 
The  officer  in  charge  of  the 
custom  house  is  called  the  Col¬ 
lector  of  Customs ,  and  a  list  of 
the  rates  of  duties  to  be  col¬ 
lected,  is  called  a  Tariff. 

Duties  are  Specific  or  Ad 
Valorem. 


268.  Specific  Duties  are  customs  assessed  on  the 
quantity  of  goods  imported,  without  reference  to  their 
value. 

In  specific  duties  an  allowance  was  formerly  made  (1)  for  waste, 
called  Draft;  (2)  for  the  weight  of  box,  cask,  etc.,  called  Tare  or  Tret; 
(3)  for  waste  of  liquids,  called  Leakage;  and  (4)  for  the  breakage  of 
bottles,  called  Breakage.  It  is  now  the  custom  to  count,  measure,  and 
weigh  imported  goods,  and  to  assess  the  duty  on  the  amounts  thus 
found. 

269.  Ad  Valorem  Duties  are  customs  assessed  on 
the  cost  of  goods  in  the  country  from  which  they  are  im¬ 
ported. 

The  cost  of  imported  goods  is  shown  by  an  Invoice  or  Manifest, 
and  when  the  currency  of  the  country  from  which  goods  are  im¬ 
ported  has  a  depreciated  value,  the  amount  of  depreciation  is  stated 
.  in  a  consular  certificate,  attached  to  the  invoice.  When  the  owner 
or  consignee  can  not  exhibit  an  invoice  of  goods  at  the  custom  house, 
their  value  is  determined  by  appraisement. 


CUSTOMS. 


169 


WRITTEN  PROBLEMS. 

1.  What  is  the  duty,  at  5  cts.  a  pound,  on  65  casks  of 
raisins,  gross  weight  115  lb.  each,  tare  12  %  ? 

2.  What  is  the  duty,  at  25  cts.  a  pound,  on  1240  chests 
of  tea,  gross  weight  120  lb.  each,  tare  10  %  ? 

3.  What  is  the  duty,  at  5  cts.  a  pound,  on  340  sacks  of 
coffee,  250  lb.  gross  each,  tare  5  %  ? 

4.  What  is  the  duty,  at  1^-  cts.  a  pound,  on  240  tons  of 
bar  iron,  draft  5  %  ? 

Note. — Irf  custom  house  computations  a  cwt.  =  112  lb. 

5.  A  merchant  imported  from  Havana  225  hogsheads  of 
sugar,  475  lb.  gross  each,  tare  12 J  %  ;  and  120  hogsheads 
of  molasses,  126  gal.  each,  leakage  2  % ;  what  was  the  duty, 
at  3  cts.  a  lb.  for  sugar,  and  8  cts.  a  gal.  for  molasses  ? 

6.  A  merchant  imported  a  lot  of  silks,  invoiced  at  $45360: 
what  was  the  duty,  at  50  %  ad  valorem  ? 

7.  A  merchant  imported  1450  yards  of  broadcloth,  in¬ 
voiced  at  $2.15  a  yd.;  3240  yards  Brussels  carpeting,  in¬ 
voiced  at  $1.60  a  yd. ;  and  480  yards  of  silk,  invoiced  at 
$2.85:  how  much  was  the  duty,  at  35%  for  the  woolen 
goods,  and  50  %  for  the  silk  ? 

8.  The  duty  on  1250  yards  of  silk,  at  40  %  ad  valorem, 
was  $1100:  what  was  the  invoice  price  a  yard?  For  how 
much  a  yard  must  the  importer  sell  the  silk  to  clear  20  %  ? 

FORMULAS  AND  RULES. 

270.  Formulas. — 1.  Specific  duty = net  quantity  X.  rate. 

2.  Ad  val.  duty = net  inv.  price  X  cate  % . 

3.  Net  inv.  price  =  ad  val.  duty  rate  % . 

271.  Rules. — 1.  To  find  specific  duty,  Midtiply  the  num¬ 
ber,  denoting  the  net  quantity  of  the  goods,  by  the  duty  on 
one. 

2.  To  find  ad  valorem  duties,  Multiply  the  invoice  price  less 
deductions  allowed,  by  the  rate  per  cent  expressed  decimally. 

C.Ar.— 15. 


170 


COMPLETE  ARITHMETIC. 


BANKRUPTCY. 

272.  A  Bankrupt  is  a  person  who  fails  in  business 
and  has  not  property  enough  to  pay  all  his  debts.  A  bank¬ 
rupt  is  also  called  an  Insolvent. 

Note. — The  term  bankrupt  is  strictly  applicable  only  to  a  trader, 
while  the  term  insolvent  applies  to  any  person  who  is  unable  to  pay 
his  debts. 

273.  Bankruptcy  is  a  failure  in  business,  with  ina¬ 
bility  to  pay  all  debts. 

274.  An  Assignment  is  the  transfer  of  the  property 
of  a  bankrupt  to  certain  persons,  called  Assignees,  in  whom  it 
is  vested  for  the  benefit  of  the  creditors. 

Note. — It  is  the  duty  of  assignees  to  convert  the  property  into 
money  and  divide  the  proceeds,  after  deducting  expenses,  among  the 
creditors. 

275.  The  property  of  a  bankrupt  or  insolvent  is  called 
his  Assets,  and  the  amount  of  his  indebtedness  is  called  his 
Liabilities.  The  assets  less  the  expense  of  settling  are  the 
Net  Proceeds. 

WRITTEN  PROBLEMS. 

1.  A  merchant  failed  in  business,  owing  $15750,  and  his 
assets  amount  to  $10515:  what  per  cent  of  his  liabilities 
can  he  pay,  allowing  $750  for  expense  of  settling. 

Process.  Since  the  net  proceeds  of 

$10515  —  $750  =  $9765,  net  proceeds.  his  affts  are  but  -62  of  his 
$9765  -r-  $15750  =  .62,  or  62  %  liabilities,  he  can  pay  but 

62$?,  or  62  cts.  on  a  dollar. 

2.  In  the  above  case  of  bankruptcy  there  are  four  cred¬ 
itors,  whose  claims  are  respectively,  $3580,  $4635,  $5300, 
and  $2235 :  how  much  will  each  receive  ? 

3.  Smith,  Jones  &  Co.  have  become  insolvent,  owing  A 
$3500,  B  $1250,  C  $3750,  D  $1000,  and  E  $2500;  their 
assets  amount  to  $7150,  and  the  expense  of  settling  will 
be  $550:  what  per  cent  of  their  liabilities  can  they  pay? 
What  will  each  creditor  receive? 


INTEREST. 


171 


4.  A  dry  goods  merchant  failed,  with  liabilities  amount¬ 
ing  to  $25000;  his  assets  are:  goods  $9500,  building  and 
lot  $5400,  and  bills  collectible  $2100 ;  and  the  expense  of 
settling  will  be  5  %  of  the  assets.  How  many  cents  on  the 
dollar  can  he  pay? 

FORMULAS  AND  RULES. 

276.  Formulas. — 1.  Bate  %  =  net  'proceeds  ■—  liabilities. 

2.  Dividend  =  claim  X  TcAe  %. 

277.  Rules. — 1.  To  find  what  per  cent  of  his  liabilities 
a  bankrupt  can  pay,  Divide  the  net  proceeds  of  his  assets  by 
the  amount  of  his  liabilities ,  and  the  quotient  expressed  in  hun¬ 
dredths  will  be  the  rate  per  cent. 

2.  To  find  the  dividends  of  creditors  in  a  case  of  bank¬ 
ruptcy,  Multiply  the  several  claims  of  creditors  by  the  rate  per 
cent  which  the  net  proceeds  of  the  assets  will  pay. 

Note. — It  is  more  common  to  find  how  many  cents  on  the  dollar 
the  net  proceeds  will  pay;  the  process  is  the  same. 


INTEREST. 

PRELIMINARY  DEFINITIONS. 

278.  Interest  is  compensation  for  the  use  of  money. 

279.  The  Principal  is  the  sum  of  money  for  the  use 
of  which  interest  is  paid. 

280.  The  Amount  is  the  sum  of  the  principal  and  in¬ 
terest. 

281.  The  Pate  of  Interest  is  the  number  of  hun¬ 
dredths  of  the  principal  paid  for  its  use  one  year.  . 

282.  The  rate  of  interest  fixed  by  law  is  called  the  legal 
rate;  and  any  rate  of  interest  higher  than  the  legal  rate  is 
usury. 

The  legal  rate  of  interest  in  most  of  the  states,  and  on  debts  due 
the  United  States,  is  6.  In  several  states  a  rate  higher  than  the 
legal  rate  is  allowed,  when  so  stipulated  in  the  contract.  (Art.  438). 


172 


COMPLETE  ARITHMETIC. 


283.  Simple  Interest  is  interest  on  the  principal  only. 

Interest  considers  the  element  of  time ,  in  which  respect  it  differs 
from  the  previous  applications  of  percentage.  For  periods  of  time 
greater  or  less  than  one  year,  the  interest  is  proportionally  greater  or 
less  than  the  interest  for  one  year. 


GENERAL  METHOD  OF  COMPUTING  INTEREST. 

MENTAL  PROBLEMS. 

1.  When  money  is  loaned  at  6  %  a  year,  what  part  of 
the  principal  equals  the  interest  for  one  year? 

Ans. — Since  6  %  —  .06,  the  interest  for  one  year  equals  .06  of  the 
principal. 

2.  What  part  of  the  principal  equals  the  interest,  when 
money  is  loaned  at  5  %  ?  At  8  %  ?  At  10  %  ? 

3.  What  part  of  the  principal  equals  the  interest,  when 
money  is  loaned  at  4  %  ?  At  %  ?  At  7-J-  %  ? 

4.  What  is  the  interest  of  $50  for  one  year  at  6  %  ? 

Solution. — Since  the  interest  for  one  year,  at  6  %,  equals  .06  of 
the  principal,  the  interest  of  $50  for  one  year  equals  .06  of  $50,  which 
is  $3. 

5.  What  is  the  interest  of  $400  for  one  year  at  7  %  ? 
At  8  %  ?  At  9  %  ?  At  10  %  ? 

6.  What  is  the  interest  of  $650  for  one  year,  at  4  %  ? 
At  6  %  ?  At  8  %  ? 

7.  What  is  the  interest  of  $120  for  one  year,  at  5  %  ? 
At  5^%?  At  10  %  ? 

8.  What  is  the  interest  of  $250  for  3  years,  at  6  %  ? 

Solution. — The  interest  of  $250  for  1  year,  at  6%,  is  $15,  and 
since  the  interest  for  1  year  is  $15,  the  interest  for  3  years  is  3  times 
$15,  which  is  $45. 

9.  At  7  % ,  what  is  the  interest  of  $300  for  4  years  ? 
For  5  years?  10  years? 

10.  At  8%,  what  is  the  interest  of  $150  for  2  years? 
4\  years  ?  5^-  years  ?  8  years  ? 


INTEREST. 


173 


11.  At  44%,  what  is  the  interest  of  $200  for  3  years? 

41  years  ?  6^  years  ? 

12.  At  10  %,  what  is  the  interest  of  $25  for  6  years?  12 
years  ?  8f  years  ? 

13.  What  is  the  interest  of  $70  for  2  years  and  4  months, 
at  5  %  ? 

Suggestion. — 4  months  =  |  of  a  year,  and  2  yr.  4  mo.  =  yrs. 

14.  At  4%,  what  is  the  interest  of  $15  for  3  years  3 
months?  For  5  years  6  months? 

15.  At  7  %,  what  is  the  interest  of  $30  for  2  years  4 
months?  3  years  2  months? 

16.  At  6  %,  what  is  the  interest  of  $50  for  4  years  2 
months?  6  years  10  months? 

17.  What  is  the  interest  of  $10  for  4  years  6  months,  at 

4  %  ?  At  6  %  ?  At  9  %  ?  At  10  %  ? 

18.  What  is  the  interest  of  $500  for  3  years  2  months,  at 

5  %  ?  At  8  %  ?  At  12  %  ? 

WRITTEN  PROBLEMS. 

19.  What  is  the  interest  of  $145.60  for  5  years  10  months, 
at  5  %  ? 

Process. 

$145.60 
_ 1)5 

$7.2800  =  Int.for  1  year. 

HA 

°6 

36.40 

6.07 

$42.47  =  Int.  for  5  yr.  10  mo. 

20.  What  is  the  interest  of  $273.45  for  8  years  3  months, 
at  %  ?  At  10  %  ? 

What  is  the  interest  of 

21.  $65.30  for  1  yr.  3  mo.,  at  6  %  ?  At  8  %  ? 

22.  $640.58  for  4  yr.  11  mo.,  at  5  %  ?  At  10  %  ? 

23.  $1000  for  1  yr.  1  mo.,  at  3-|-  %  ?  At  7.3  %  ? 

24.  $85,  at  7  % ,  for  3  yr.  7  mo.  ?  For  9  months  ? 


174 


COMPLETE  ARITHMETIC. 


25.  $38.10,  at  9  %,  for  6  yr.  5  mo.  ?  For  3  yr.  10  mo.? 

26.  $84.75  for  2  yr.  5  mo.  21  da.,  at  8  %  ? 

Suggestion. — Reduce  the  5  mo.  21  da.  to  the  decimal  of  a  yeai. 
(Art.  171,  N.  2.)  21  da.  =  .7  mo.,  and  5.7  mo.— .475  yr.  Hence, 
2  yr.  5  mo.  21  da.  =2.475  yr. 

27.  $208.44  for  7  yr.  8  mo.  15  da.,  at  5  %? 

28.  $356.75  for  5  yr.  10  mo.  24  da.,  at  6J  %  ? 

29.  $184.80  for  1  yr.  1  mo.  10  da.  (1^  yr.),  at  9%  ? 

30.  $321.70  for  4  yr.  3mo.  27  da.,  at  12%  %  ? 

What  is  the  amount  of 

31.  $60.85  for  10  yr.  10  mo.  10  da.,  at  10  %  ? 

32.  $740.10  for  1  yr.  1  mo.  18  da.,  at  8  %  ? 

33.  $1.40  for  7  yr.  11  mo.  21  da.,  at  7|  %  ? 

34.  $121.75  for  3  yr.  18  da.,  at  12  %  ? 

35.  $80.65  for  1  yr.  6  mo.  12  da.,  at  10^  %  ? 

36.  What  is  the  interest  of  $356.50  for  3  yr.  9  mo.  25 
da.,  at  8  %  ? 

Process  by  Aliquot  Parts  for  Days. 

$356.50 
_ .08 

12  )  $28.5200  X  3  =  $85,560  Int.  for  3  yrs. 

(Int  fort'  1  mo.)  $2.3766X9=  21.389  “  “  9  mo. 

15  da.  =  |  mo.  1.188  “  “  15  da. 

10  da.  =  j  mo.  .792  “  “  10  da. _ 

$108,929  Int.  for  3  yr.  9  mo.  25  da. 

What  is  the  interest  of 

37.  $84.66  for  5  yr.  7  mo.  20  da.,  at  5  %? 

38.  $4000  for  10  yr.  10  mo.  10  da.,  at  15  %? 

39.  $1262.70  for  11  mo.  27  da.,  at  1\  %  ? 

40.  $504.08  for  3  yr.  1  mo.  1  da.,  at  10  %  ? 

41.  $3084.90  for  7  mo.  22  da.,  at  12  %  ? 

42.  $2016.05  for  1  yr.  1  mo.  29  da.,  at  8  %  ? 

43.  What  is  the  amount  of  $262.75  for  1  yr.  5  mo.  19 
da. ,  at  6  %  ?  At  7  %  ?  At  9  %  ? 

44.  What  is  the  amount  of  $192.60  for  2  yr.  2  mo.  2 
da.,  at  5  %  ?  At  10  %  ?  At  12  %  ? 


INTEREST. 


175 


45.  A  man  borrowed  $60  May  10,  1864,  and  paid  it 
March  4,  1866,  with  interest  at  6  % :  what  amount  did  he 
pay? 

Suggestion. — Find  the  difference  of  time  by  compound  subtrac¬ 
tion. 

46.  What  is  the  interest  of  $15.80,  from  Oct.  23,  1855, 
to  Apr.  12,  1859,  at  8  %  ? 

47.  A  note  of  $565.80,  dated  June  3,  1864,  was  paid  Nov. 
28,  1869,  with  interest  at  8  %  :  what  was  the  amount  paid? 


PRINCIPLE,  FORMULA,  AND  RULE. 

284.  Principle. — The  principal  multiplied  by  the  rate  per 
cent  equals  the  interest  for  one  year. 

285.  Formula. — Interest  =  principal  X  rate%  X  ti 

286.  Rule. — To  find  the  interest  of  any  sum  of  money 
for  any  time,  at  any  rate  per  cent,  1.  Multiply  the  principal 
by  the  rate  per  cent ,  expressed  decimally ,  and  multiply  this  prod¬ 
uct  by  the  time  in  years  and  the  fraction  of  a  year.  Or, 

2.  Multiply  the  interest  for  one  year  by  the  number  of  years, 
and  Y2  of  it  by  the  number  of  months,  and  find  the  interest  for 
days  by  aliquot  parts.  The  sum  of  the  several  results  will  be  the 
interest  for  the  given  time. 

Note. — In  solving  the  majority  of  problems  in  interest,  the  re¬ 
duction  of  the  months  and  days  to  the  decimal  of  a  year  will  be 
found  as  brief  as  the  method  by  aliquot  parts.  Those  who  prefer  to 
use  aliquot  parts,  will  find  the  method  given  above  briefer  than  the 
one  generally  used. 


SIX  PER  CENT  METHOD. 

MENTAL  PROBLEMS. 

1.  What  is  the  interest  of  $1  for  2  years,  at  6  %  ? 

Solution. — Th«  interest  of  $1  for  1  year  at  6%  is  .06  of  $1, 
which  is  6  cents,  and  the  interest  for  2  years  is  2  times  6  cents,  which 
is  12  cents. 


176 


COMPLETE  ARITHMETIC. 


2.  What  is  the  interest  of  $1,  at  6  %,  for  3  years?  For 
8  years?  12  years?  15  years? 

3.  What  is  the  interest  of  $1,  at  6  %,  for  5  years?  For 
10^  years?  16|  years? 

4.  What  is  the  interest  of  $1  for  1  month,  at  6  %  ?  For 
3  months? 

Solution. — Since  the  interest  of  $1  for  1  year,  at  6%,  is  6  cents, 
the  interest  for  1  month  is  of  6  cents,  which  is  5  viills,  and  the 
interest  for  3  months  is  3  times  5  mills,  which  is  15  mills. 

5.  What  is  the  interest  of  $1,  at  6  %,  for  4  months?  6 

months  ?  8  months  ?  10  months  ? 

6.  What  is  the  interest  of  $1,  at  6  %,  for  5  months?  7 
months?  9  months?  11  months? 

At  6  %,  what  is  the  interest  of 

7.  $1  for  1  year  2  months  ?  2  yr.  4  mo.  ?  3  yr.  6  mo.  ? 

8.  $1  for  4  years  5  months  ?  6  yr.  7  mo.  ?  5  yr.  9  mo.  ? 

9.  $1  for  2  yr.  11  mo.  ?  3  yr.  9  mo.  ?  10  yr.  10  mo.  ? 

10.  What  is  the  interest  of  $1,  at  6  %,  for  1  day?  For 
6  days? 

Solution. — Since  the  interest  of  $1  for  30  days,  at  6%,  is  5  mills, 
the  interest  for  1  day,  is  -fa  of  5  mills,  which  is  £  of  1  mill,  and  the 
interest  for  6  days  is  6  times  ^  of  1  mill,  which  is  1  mill. 

11.  What  is  the  interest  of  $1,  at  6  %,  for  12  days?  For 
18  days?  For  24  days? 

12.  What  is  the  interest  of  $1,  at  6%,  for  9  days?  15 
days?  21  days?  27  days? 

13.  What  is  the  interest  of  $1,  at  6  %,  for  10  days?  20 
days?  14  days?  22  days? 

14.  What  is  the  interest  of  SI,  at  6  %,  for  7  days?  11 
days?  17  days?  23  days?  25  days? 

15.  What  is  the  interest  of  SI,  at  6%,  for  2  months  12 
days?  4  mo.  18  da.?  5  mo.  6  da.? 

At  6  % ,  what  is  the  interest  of 

16.  SI  for  6  mo.  9  da?  8  mo.  15  da.?  10  mo.  21  da.? 

17.  SI  for  5  mo.  13  da.  ?  3  mo.  22  da.  ?  7  mo.  25  da.  ? 


INTEREST. 


177 


18.  $1  for  4  mo.  16  da.  ?  6  mo.  29  da.?  10  mo.  10  da.  ? 

19.  $1  for  6  mo.  16  da.  ?  9  mo.  28  da.?  1]  mo.  11  da.? 

20.  $1  for  2  yr.  8  mo.  12  da.  ?  4  yr.  5  mo.  18  da.  ? 

21.  $1  for  3  yr.  3  mo.  24  da.  ?  5  yr.  6  mo.  6  da.  ? 

22.  At  6%,  what  is  the  interest  of  $1  for  1  year?  For 
1  month?  2  months?  For  1  day?  6  days? 

23.  If  the  interest  of  a  certain  principal  is  $12,  at  6  %, 
what  would  be  the  interest  of  it  at  7  %  ?  At  8  %  ? 

Suggestion. —  7%  is  !  more  than  6%;  and  8%  is  !  more  than  6%. 

If  the  interest  of  a  certain  principal,  at  6  %,  is  $36,  what 
would  be  the  interest  of  it 

24.  At  12  %  ?  At  15  %  ?  At  18  %  ?  At  21  %  ? 

25.  At  3%?  At  4%?  At4i%?  At  5%? 

26.  At  10%  ?  At  11  %?  At  13  %  ?  At  14  %  ? 


WRITTEN  PROBLEMS. 

27.  What  is  the  interest  of  $245.60  for  2  yr.  7  mo. 
21  da.,  at  6  %  ? 


Process. 

Int.  of  $1  =  $.158| 

$245.60 
_ .158! 

196480 

122800 

24560 

12280 

$38.9276,  Int. 


Since  the  interest  of  $1  for  2  yr.  7  mo. 
21  da.,  at  6%,  is  $.158!,  or  *168!  of  $1,  the  in¬ 
terest  of  $245.60  will  be  .158!  °f  $245.60,  which 
is  $38,928.  The  interest  is  as  many  thousandths 
of  the  principal,  as  the  interest  of  $1  is  thou¬ 
sandths  of  $1. 


28.  What  is  the  interest  of  $245.60  for  2  yr.  7  mo. 
21  da.,  at  9%?  At  11  %  ? 


Process. 


Process. 


$38,928,  Int.  at  6  fa. 

19.464,  “  “  3%. 
$58,392, 


$38,928, 

Int. 

at 

6%. 

19.464, 

U 

H 

3  %. 

12.976, 

u 

a 

2%. 

$71,308, 

n 

(t 

11%. 

It  u 


178 


COMPLETE  ARITHMETIC. 


29.  What  is  the  interest  of  $508.09  for  3  yr.  3  mo. 
15  da.,  at  6%?  At  5  %  ?  At  7  %  ? 

What  is  the  interest  of 

30.  $540  for  10  mo.  24  da.,  at  6  %?  At  8  %? 

31.  $327.50  for  1  yr.  3  mo.  6  da.,  at  7  %  ?  At  10  %  ? 

32.  $142.64  for  2  yr.  15  da.,  at  4  %  ?  At  4J  %  ? 

33.  $3008.75  for  4  yr.  1  mo.  20  da.,  at  5  %  ?  At  9  %  ? 

34.  $622.40  for  9  mo.  29  da.,  at  12  %?  At  15%  ? 

What  is-  the  amount  of 

35.  $804.25  for  1  yr.  5  mo.  10  da.,  at8%?  At  7J  %  ? 

36.  $112.40  for  11  mo.  21  da.,  at  5£%?  At  6£%? 

37.  $2000  for  1  yr.  1  mo.  1  da.,  at  8  %  ?  At  11  %  ? 

38.  $5.90  for  3  yr.  3  mo.  3  da.,  at  3  %  ?  At  12  %  ? 

39.  $16.50  for  2  yr.  2  mo.  2  da.,  at  6  %  ?  At  7-J-  %  ? 

40.  $50.30  for  3  yr.  3  mo.  3  da.,  at  8  %?  At  5  %  ? 

41.  $200  for  4  yr.  4  mo.  4  da.,  at  4  %  ?  At  10  %  ? 

42.  What  is  the  interest  of  $108.60,  from  Sept.  12,  1866, 
to  May  6,  1870,  at  6  %  ?  At  8  %  ? 

43.  A  debt  of  $40.50  was  paid  May  21,  1870,  with  in¬ 
terest,  at  6%,  from  Nov.  9,  1864:  what  was  the  amount 
paid  ? 

44.  A  note  of  $350,  dated  Oct.  17,  1865,  was  paid  Apr. 
11,  1868,  with  interest  at  7  % :  what  was  the  amount  paid? 

45.  A  note  of  $150.75,  dated  June  15,  1867,  was  paid 
Jan.  1,  1870,  with  interest  at  5  %  :  what  was  the  amount 
paid  ? 

46.  A  note  of  $1250,  dated  July  5,  1868,  was  paid 
June  1,  1870,  with  interest  at  8  % :  what  was  the  amount 
paid  ? 

47.  A  note  of  $87.50,  dated  Aug.  8,  1867,  and  bearing 
interest  at  10  %,  was  paid  March  25,  1869  :  what  was  the 
amount  paid? 

48.  A  note  of  $65.80,  dated  Feb.  20,  1868,  and  bearing 
interest  at  7%,  was  paid  June  25,  1870:  what  was  the 
amount  paid  ? 


INTEREST. 


179 


FORMULA  AND  RULES. 

287.  Formula. — Int.  at  6%  = principal  X  int.  of  $1  at  6%. 

288.  Kules. — 1.  To  compute  interest  at  6%,  Find  the 
interest  of  $  1  for  the  given  time ,  by  taking  six  times  as  many 
cents  as  there  are  years ,  one-half  as  many  cents  as  there  are 
months ,  and  one-sixth  as  many  mills  as  there  are  days;  and 
then  multiply  the  principal  by  the  abstract  decimal  which  corre¬ 
sponds  to  the  interest  of  $1  thus  found. 

2.  To  compute  interest  at  any  other  rate  than  6%,  Find 
the  interest  at  6  % ,  and  then  increase  or  diminish  this  interest  by 
such  a  part  of  itself  as  will  give  the  interest  at  the  given  rate. 


METHOD  BY  DAYS. 

289.  When  the  time  is  short,  it  is  the  custom  of  bankers 
and  other  business  men  to  compute  interest  for  the  actual 
number  of  days  included  in  the  time,  each  day  being  con¬ 
sidered  as  of  a  year. 


WRITTEN  PROBLEMS. 


1.  What  is  the  interest  of  $80.60  from  March  15th  to 
June  10th,  at  6  %  ? 


Process  : 

In  March  16  days.  $80.60 

“  Apr.  30  “  .014| 

“  May  31  “  32240 

“  June  10  tl  8060 

6  )  87  days.  4030 

~  14£  $1.16870 


Allowing  360  days  to  a  year, 
the  interest  for  87  days  is  of 
the  interest  for  one  year,  and  the 
interest  for  1  year,  at  6%,  is  rfo’ 
of  the  principal.  Hence,  the  in¬ 
terest  for  87  days  is  of  7 
of  the  principal,  which  is 
of  the  principal.  But  =  % 


of  Hence,  the  interest,  at  6  %,  equals  one-sixth  as  many  thou¬ 

sandths  of  the  principal  as  there  are  days  in  the  time. 


Note. — This  explanation  may  be  preferred :  Allowing  360  days  to 
the  year,  the  interest  of  $1  for  1  day,  at  6%,  is  of  6  cents,  or  60 
mills,  which  is  %  of  a  mill ,  and,  hence,  the  interest  of  $1  for  any  num¬ 
ber  of  days^'s  |  of  as  many  mills  as  there  are  days.  Having  found  the 
interest  of  $1,  the  interest  of  any  principal  is  found  as  in  the  preced¬ 
ing  article. 


180 


COMPLETE  ARITHMETIC. 


2.  What  is  the  interest  of  $125.80  from  July  5th  to 
Oct.  23d,  at  6  %  ?  At  8  %  ? 

3.  What  is  the  amount  of  $25.25  from  Oct.  30,  1869, 
to  Feb.  1,  1870,  at  6%?  At  7£%? 

4.  What  is  the  amount  of  $65.80  from  Dec.  28,  1867, 
to  Mch.  15,  1868,  at  5  %  ?  At  10  %  ? 

5.  What  is  the  amount  of  $75.40  from  Jan.  13,  1869, 
to  June  15,  1869,  at  6%?  At  7  %  ? 

6.  What  is  the  amount  of  $120  from  Mch.  15,  1870,  to 
July  4,  1870,  at  7  %  ?  At  9  %  ? 

7.  A  note  of  $420,  dated  Jan.  25,  1860,  was  paid  Apr. 
16,  1860,  with  interest  at  8  % :  what  was  the  amount? 

8.  A  man  borrowed  $150,  June  6th,  and  paid  it,  with 
interest  at  7  Sept.  24th:  how  much  did  he  pay? 

9.  A  note  of  $80,  dated  Jan.  15,  1868,  was  paid  June 
21,  1868,  with  interest  at  8  %  :  what  was  the  amount? 

10.  A  note  of  $150,  dated  Mch.  30,  1870,  was  paid  July 
4,  1870,  with  interest  at  6  %  :  what  was  the  amount? 

11.  A  note  of  $500,  dated  May  12,  1869,  and  bearing 
interest  at  7  %,  was  paid  July  24,  1869:  what  was  the 
amount? 


FORMULA  AND  RULES. 

290.  F ormula. — Interest  at  6  %  — principal  X  days  6000. 

291.  Rules. — 1.  To  compute  interest  for  days  at  6  %, 
Multiply  the  principal  hy  one  sixth  of  as  many  thousandths  as 
there  are  days  in  the  time. 

2.  To  compute  interest  for  days  at  any  %,  Find  the  in¬ 
terest  at  6  % ,  and  then  increase  or  diminish  this  interest  by  such 
a  part  of  itself  as  the  given  rate  is  greater  or  less  than  6. 

Notes. — 1.  Since  the  common  year  consists  of  365  days,  instead 
of  360,  the  true  interest  for  360  days  is  or  fa  of  the  interest  for 
a  year;  whereas,  by  the  above  method,  the  interest  for  360  days  equals 
the  interest  for  a  year.  Hence,  the  true  interest  for  any  number  of 
days  in  a  common  year  is  fa  less  than  the  interest  found  by  the  above 
rule ;  in  leap  year  the  true  interest  is  fa  less  than  the  interest  thus 
found.  An  accurate  rule  for  computing  interest  for  days  is,  to  take 


PARTIAL  PAYMENTS. 


181 


as  many  365 ths,  and  in  leap  year  as  many  366ihs,  of  the  interest  for  one 
year  as  there  are  days  in  the  time.  • 

2.  In  Great  Britain,  a  day’s  interest  is  made  of  a  year’s  inter¬ 
est,  and  the  same  rule  is  adopted  by  the  United  States  Government  in 
computing  interest  upon  bonds,  etc.  The  convenience  of  the  method 
which  allows  360  days  to  the  year  has  secured  its  very  general  adop¬ 
tion  by  the  business  men  of  the  country,  and  in  several  states  it  is 
sanctioned  by  law. 

3.  There  are  three  methods  of  finding  the  time  between  two  dates, 
to-wit:  1.  By  compound  subtraction,  allowing  30  days  to  the  month.  2. 
By  finding  the  number  of  calendar  months  from  the  first  date  to  the  corre¬ 
sponding  day  of  the  month  of  the  second  date,  and  then  counting  the  actual 
number  of  days  left.  3.  By  counting  the  actual  number  of  days  between 
the  two  dates.  The  third  method  is  strictly  accurate,  and  is  generally 
used  in  finding  the  time  of  “  short  paper.”  The  number  of  days  may 
be  found  from  “  Time  Tables,”  which  give  the  exact  number  of  days 
between  any  two  dates  less  than  a  year  apart. 


PARTIAL  PAYMENTS. 

292.  When  partial  payments  have  been  made  on  notes  and 
other  obligations,  the  interest  is  computed  by  the  following 
rule,  which,  having  been  adopted  by  the  Supreme  Court  of 
the  United  States,  is  called  the 

UNITED  STATES  RULE. 

When  'partial  payments  have  been  made,  apply  the  payment, 
in  the  first  place,  to  the  discharge  of  the  interest  then  due.  If 
the  payment  exceeds  the  interest,  the  surplus  goes  toward  dis¬ 
charging  the  principal,  and  the  subsequent  interest  is  to  be  com¬ 
puted  on  the  balance  of  principal  remaining  due. 

If  the  payment  be  less  than  the  interest,  the  surplus  of  inter¬ 
est  must  not  be  taken  to  augment  the  principal,  but  interest  con¬ 
tinues  on  the  former  principal  until  the  period  when  the  pay¬ 
ments,  taken  together,  exceed  the  interest  due,  and  then  the  sur¬ 
plus  is  to  be  applied  toward  discharging  the  principal,  and  the 
interest  is  to  be  computed  on  the  balance,  as  aforesaid. 

293.  This  rule  requires,  first,  that  payments  be  applied 
to  the  discharge  of  interest  then  due ;  and,  secondly,  that 
no  unpaid  interest  be  added  to  the  principal  to  draw  inter¬ 
est.  Interest  accrues  only  on  the  unpaid  principal. 


182 


COMPLETE  ARITHMETIC. 


*  PROBLEMS. 

1.  A  note  of  $650,  dated  May  20,  1866,  and  drawing  in¬ 
terest  at  6  %,  had  payments  indorsed  upon  it  as  follows: 

Sept.  2,  1866,  $25.  March  2,  1867,  $150. 

Dec.  20,  1866,  $10.  July  8,  1867,  $200. 

What  was  the  amount  due  Nov.  11,  1867? 


Process. 


$650 

.017 

1866 

9  2 

4550 

650 

1866 

5  20 

$11,050 

ls£  interest. 

3  mo.  12  da. 

.017 

650. 

1866 

1866 

$25. 

12  20 

9  2 

661.05 

25.00 

$636.05 

.018 

lsi  payment. 

2 d  principal. 

3  mo.  18  da. 

$10. 

.018 

508840 

63605 

$10  $11.44890 

2d  interest. 

1867 

1866 

3  2 

12  20 

$636.05 

.012 

od  principal. 

2  mo.  12  da. 
$150. 

.012 

$150  $7.63260 

11.4489 
636.05 

3 d  interest. 

2d  interest. 

1867 

1867 

7  8 

3  2 

$655.1315 

160.00 

$495.1315 

.021 

2d  +  3d  payment. 
4 th  principal. 

4  mo.  6  da. 

.021 

$200. 

4951315 

9902630 

1867 

1867 

11  11 

7  8 

$10.3977615 

495.1315 

4 th  interest. 

4  mo.  3  da. 

.0205 

$505.5293 

200.00  4 th  payment. 

$305.5293  5th  principal. 

.0205 

15276465 

6110586 

$6.26335065  5th  interest. 

305.5293 

$311.7926,  Amount  due  Nov.  11,  1867. 


PARTIAL  PAYMENTS. 


183 


The  first  step  is  to  find  the  difference  of  time  between  each  two 
consecutive  dates,  and  form  the  corresponding  decimal  multipliers 
by  the  six  per  cent  method  (Art.  288).  The  payments  may  be 
written  below.  This  preparation  will  lessen  the  liability  of  error  in 
the  calculation. 

Since  the  1st  payment  is  greater  than  the  1st  interest,  form  the 
amount  and  subtract  therefrom  the  payment.  The  difference  is  the 
2d  principal.  Find  the  2d  interest. 

Since  the  2d  payment  is  less  than  the  2d  interest,  let  the  interest 
stand,  drawing  a  double  line  beneath  it,  and  bringing  down  the  2d 
principal  for  a  3d  principal.  Find  the  3d  interest. 

Since  the  sum  of  the  2d  and  3d  payments  is  greater  than  the  sum 
of  the  2d  and  3d  interests,  form  the  amount  and  subtract  therefrom 
the  sum  of  the  2d  and  3d  payments.  The  difference  is  the  4th  prin¬ 
cipal.  Find  the  4th  interest. 

Since  the  4th  payment  is  greater  than  the  4th  interest,  form  the 
amount  and  subtract  therefrom  the  4th  payment.  Compute  the  in¬ 
terest  on  the  difference,  the  5th  principal,  to  the  last  date,  and  form 
the  amount,  which  is  the  sum  then  due. 

Notes. — 1.  Sometimes  an  estimate  of  the  interest  may  be  made  men¬ 
tally  with  sufficient  accuracy  to  determine  whether  it  is  greater  or  less 
than  the  payment.  If  greater,  the  sum  of  the  two  or  more  decimal 
multipliers,  can  be  used  for  a  multiplier.  Instead  of  multiplying 
by  .018  and  .012  above,  their  sum,  or  .03,  might  have  been  used. 
When  the  rate  is  other  than  6%,  the  several  interests  should  be  in¬ 
creased  or  diminished,  as  the  rate  may  require,  before  forming  the 
amounts. 

2.  The  above  rule  is  generally  used  when  the  time  between  the 
date  of  the  note  and  its  payment  exceeds  one  year. 

2.  A  note  of  $600,  dated  June  10,  1867,  had  indorse¬ 
ments  as  follows:  Dec.  4,  1867,  $50;  Mch.  25,  1868,  $12; 
July  9,  1868,  $75.  How  much  was  due  Oct.  15,  1868,  at 
6  %  interest? 

3.  A  note  of  $1000,  dated  Apr.  10,  1864,  wras  indorsed 
as  follows:  Nov.  10,  1865,  $80.50;  July  5,  1866,  $100; 
Jan.  10,  1867,  $450.80;  Oct.  1,  1869,  $500.  What  was 
due  Jan.  1,  1870,  at  7  %  interest? 

4.  A  note  of  $450,  dated  July  4,  1868,  was  indorsed  as 
follows:  Jan.  20,  1869,  $15;  June  9,  1869,  $200;  Oct.  20, 
1869,  $10.  What  was  due  Jan.  10,  1870,  at  10  %  interest? 

5.  A  note  of  $850,  dated  March  4,  1865,  had  indorse- 


184 


COMPLETE  ARITHMETIC. 


merits  as  follows:  Sept.  1,  1865,  $12;  May  4,  1866,  $10; 
Sept.  15,  1866,  $250  ;  Jan.  20,  1867,  $400.  What  was  due 
July  1,  1868,  at  6  %  interest? 

6.  A  note  of  $520,  dated  Apr.  12,  1867,  had  three  in¬ 
dorsements  as  follows:  Dec.  6,  1867,  $120;  July  9,  1868, 
$12  ;  Nov.  30,  1868,  $9.  What  was  due  May  1,  1869,  at 
9  %  interest? 

7.  $1250.  Cincinnati,  July  1,  1868. 

On  demand,  I  promise  to  pay  Peter  Smith,  or  order, 
twelve  hundred  and  fifty  dollars,  with  interest  at  %,  for 
value  received.  John  Coons. 

Indorsements:  Sept.  14,  1868,  $300;  Jan.  20,  1869,  $12; 
Oct.  20,  1869,  $20;  Nov,  8,  1869,  $500. 

What  was  due  on  the  above  note  Jan.  1,  1870? 

8.  $1000.  San  Francisco,  Apr.  10,  1867. 

For  value  received,  I  promise  to  pay  to  Wm.  Penn,  Jr., 
or  order,  thirty  days  after  date,  one  thousand  dollars,  with 
interest  at  10  %.  Gould  Dives. 

Indorsements:  July  28,  1867,  $500;  Dec.  13,  1867,  $8; 
Feb.  25,  1868,  $12;  July  7,  1868,  $125;  Oct.  3,  1868, 
$200;  Mch.  15,  1869,  $50. 

What  was  due  on  the  above  note  June  3,  1869. 

294.  When  partial  payments  are  made  on  mercantile  ac¬ 
counts,  past  due,  and  on  notes  running  a  year  or  less,  the 
interest  is  often  computed  by 

THE  MERCHANT’S  RULE. 

Compute  the  interest  on  the  principal  from  the  time  it  begins  to 
draw  interest  to  the  time  of  settlement,  and  also  on  each  paigment 
from  the  time  it  was  made  to  the  time  of  settlement. 

From  the  sum  of  the  principal  and  its  interest,  subtract  the 
sum  of  the  payments  and  their  interests,  and  the  difference  will 
be  the  balance  due. 


INTEREST. 


185 


9.  A  note  of  $800,  dated  March  12,  1869,  and  drawing 
interest  at  8  %,  was  indorsed  as  follows:  May  15,  1869 
$200;  Aug.  10,  1869,  $75;  Oct.  20,  1869,  $125.  What 
was  due  Dec.  30,  1869? 

10.  Payments  were  made  on  a  debt  of  $350,  due  Feb.  1, 
1868,  as  follows:  March  20,  1868,  $45;  May  1,  1868,  $60; 
July  5,  1868,  $80 ;  Oct.  1,  1868,  $50.  What  was  due  Nov. 
1,  1868,  at  6  %  interest? 

Notes. — 1.  There  are  several  other  rules  for  computing  interest 
when  partial  payments  are  made,  but  they  are  not  in  general  use, 
and  hence  are  omitted.  The  “  Vermont  Rule,”  long  used  in  Ver¬ 
mont,  and  the  “  Connecticut  Rule,”  used  in  Connecticut,  are  modifi¬ 
cations  of  the  United  States  Rule. 

2.  The  chief  aim  of  legislative  enactments  on  this  subject  has  been 
to  protect  the  debtor  from  paying  interest  on  interest ,  but  there  is  no 
essential  difference  between  applying  payments  to  the  discharge  of 
interest  instead  of  principal,  and  paying  interest  on  such  accrued  in¬ 
terest.  The  debtor  loses  the  use  of  so  much  of  every  payment  as  is 
applied  to  interest,  and  the  creditor  gains  the  use  of  it.  The 
“  Merchant’s  Rule  ”  is  the  only  one  that  does  not  practically  allow 
interest  on  interest. 


FIVE  PROBLEMS  IN  INTEREST. 

295.  Five  quantities  are  considered  in  interest,  and  such 
is  the  relation  between  them  that,  if  any  three  are  given, 
the  other  two  may  be  found.  These  quantities  are  the 
Principal,  Rate  Per  Gent ,  Time,  Interest,  and  Amount.  There 
are  five  classes  of  problems  of  practical  importance. 

Problem  I. 

Principal,  Rate  Per  Cent,  and.  Time  given,  to  find 
the  Interest  and  Amount. 

Note. — This  problem  has  already  been  considered.  The  follow¬ 
ing  problems  may  be  solved  by  the  pupil  by  either  of  the  preceding 
methods,  but  the  time  in  the  last  three  problems  should  be  found  by 
the  method  by  days. 

1.  What  is  the  interest  of  $12.50  for  3  yr.  1  mo.  15  da., 
at  6  %  ?  At  10  %  ? 

C.Ar.— 16. 


186 


COMPLETE  ARITHMETIC. 


2.  What  is  the  interest  of  $160.80  for  2  yr.  3  mo.  3  da., 
at  7  %  ?  At  9  %  ? 

3.  What  is  the  interest  of  $56.40  for  21  days,  at  10  %  ? 
What  is  the  amount  ? 

4.  What  is  the  interest  of  $1000  from  May  13,  1867,  to 
July  8,  1868,  at  6  %  ?  At  7|  %  ? 

5.  What  is  the  amount  of  $204.50  from  Jan.  21,  1869, 
to  Feb.  3,  1870,  at  9  %  ?  At  12  %  ? 

6.  What  is  the  interest  of  $80.25  from  May  15,  1869, 
to  Sept.  24,  1869,  at  6  %  ?  At  8  %  ? 

7.  A  note  of  $920,  dated  Nov.  12,  1869,  was  paid  Apr. 
3,  1870:  what  was  the  amount,  at  9  %  ? 

8.  A  note  of  $7.50,  dated  Apr.  20,  1870,  was  due  Oct. 
12,  1870,  with  interest  at  8  %  :  what  was  the  amount? 

296.  Formulas. — 1.  Interest  ^principal  X  rate%  X  time. 

2.  Amount  = principal  -f-  interest. 


Problem  II. 


Principal,  Interest,  and  Time  given,  to  find  the 

It  ate  Per  Cent. 


9.  The  interest  on  $540  for  8  mo.  18  da.  was  $27.09: 
what  was  the  rate  per  cent? 


Process. 

$540 

.043 

1620 

2160 

6  )  $23.220  Int.at&^o. 
$3.87  “  “  1  °/o. 

$27.09  -r-  $3.87  =  7. 


Since  the  interest  on  $540  for  8  mo. 
18  da.,  at  1%,  is  $3.87,  the  rate  %  which 
produced  $27.09  interest,  was  as  many 
times  1  %  as  $3.87  is  contained  times  in 
$27.09,  which  is  7.  Hence,  the  interest 
accrued  at  7fo. 


Note. — The  interest  at  1%  may  be  found  by  multiplying  $540  by 
l  of  .043,  which  is  .007£. 


10.  The  interest  of  $456  for  3  yr.  5  mo.  18  da.,  is 
$79.04:  what  is  the  rate  per  cent? 

11.  The  interest  of  $216  for  5  yr.  7  mo.  27  da.,  is 
$122.22:  what  is  the  rate  per  cent? 


PROBLEMS  IN  INTEREST. 


187 


12.  The  interest  of  $560  for  2  yr.  4  mo.  15  da.,  was 
$106.40:  what  was  the  rate  per  cent? 

13.  The  interest  of  $95.40  for  3  yr.  9  mo.,  is  $28.62: 
wThat  is  the  rate  per  cent? 

14.  The  interest  of  $240  from  Feb.  15,  1868,  to  Apr. 
27,  1869,  was  $23.04:  what  was  the  rate  per  cent? 

15.  The  interest  of  $252  from  Aug.  2,  1867,  to  March  9, 
1868,  was  $12.152 :  what  was  the  rate  per  cent? 

16.  A  note  of  $345.60,  dated  Feb.  5,  1863,  was  paid 
Aug.  20,  1865,  and  the  amount  was  $407,088:  what  was 
the  rate  per  cent? 


FORMULA  AND  RULE. 

297.  Formula. — Rate  =  interest  A-  (prin.  X  1%  X  time.) 

298.  Rule. — To  find  the  rate  per  cent,  Divide  the  given 
interest  by  the  interest  of  the  principal  for  the  given  time ,  at  1 
per  cent.  The  quotient  will  be  the  rate. 

Problem  III. 

Principal,  Interest,  and.  Pate  Per  Cent  given,  to 

find  the  Time. 

17.  The  interest  of  $300,  at  9%,  is  $60.75:  what  was 
the  time? 

Process. 

$300 
_ .09 

$27.00,  Int.for  1  yr. 

$60.75 -f- $27  =  2.25 
2.25  yr.  =  2  yr.  3  mo. 

18.  The  interest  of  $908,  at  3-J-  %,  was  $79.45 :  what  was 
the  time? 

19.  The  interest  of  $56.78  for  a  certain  time,  at  10%, 
was  $22.24:  what  was  the  time? 

20.  How  long  must  a  note  of  $300  run  to  give  an  amount 
of  $347.25,  at  6  %  ? 


Since  the  interest  of  $300  for  1  year, 
at  9%,  is  $27,  $300  must  be  on  interest 
as  many  years  to  produce  $60.75  inter¬ 
est,  as  $27  are  contained  times  in  $60.75, 
which  is  2.25.  Hence  the  time  is  2.25 
years,  or  2  yr.  3  mo.  (Art.  170). 


188 


COMPLETE  ARITHMETIC. 


21.  In  what  time  will  the  interest  of  $150,  at  4%,  be 

$9?  $11? 

22.  In  what  time  will  $2040  produce  $334.05  interest,  at 
5^? 

23.  In  what  time  will  any  principal  double  itself  at  4% 
interest?  At  6  %  ?  At  10  %  ? 

24.  In  what  time  will  any  principal  double  itself  at  5  % 
interest?  At  7  %  ?  At  12  %  ? 

25.  In  what  time  will  any  principal  treble  itself,  at  5  % 
interest?  At  10  %  ?  At  6  %  ? 

FORMULA  AND  RULE. 

299.  Formula.  —  Time  =  interest  ~  ( principal  X  rate  %). 

300.  Rule. — To  find  the  time,  Divide  the  given  interest  by 
the  interest  of  the  'principal  for  1  year ,  at  the  given  rate  per 
cent. 

Note. — Reduce  the  fraction  of  a  year  to  months  and  days.  If 
preferred,  the  interest  may  be  divided  by  the  interest  for  1  month,  at 
the  given  rate  per  cent. 


Problem  IV. 


Interest,  Pate  Per  Cent,  and  Time  given,  to  find 

the  [Principal. 


26.  What  principal  will  produce  $49.20  of  interest  in  1 
yr.  4  mo.  12  da.,  at  6  %  ? 


1st  Process. 

$.082  =Int.  of  $1. 

$.082  )  $49.20  ( 600 
492 

$1  X  600  =  $600.  Ans. 


Since  $1  of  principal  produces 
$.082  of  interest,  it  will  take  as 
many  times  $1  of  principal  to  pro¬ 
duce  $49.20  of  interest  as  $.082  is 
contained  times  in  $49.20,  which  is 
600.  600  times  $1  =  $600,  the  re¬ 

quired  principal.  Or, 


2d  Process. 


.082 )  $49,200  (  $600 
492 


Since  the  interest  of  $1  is  .082 
of  itself,  $49.20  is  .082  of  the  re¬ 
quired  principal.  $49.20-;-  .082  = 
$600. 


PROBLEMS  IN  INTEREST. 


189 


What  principal  will  produce 

27.  $15.24  interest  in  7  mo.  6  da.,  at  8  %  ? 

28.  $1000  interest  in  5  yr.  6  mo.  20  da.,  at  5%? 

29.  $519  interest  in  5  mo.  23  da.,  at  12  %? 

30.  What  sum  invested,  at  7  %,  will  produce  $378  inter¬ 
est  annually  ? 

31.  What  sum  invested,  at  4\  will  yield  an  annual 
income  of  $900? 

32.  What  principal  will  produce  $220  interest,  from  Oct. 
25,  1871,  to  March  7,  1872,  at  8  %  ? 

33.  What  principal  will  produce  $17.78  interest,  from 
Jan.  10,  1872,  to  March  13,  1872,  computed  by  days,  at  4%? 


FORMULA  AND  RULES. 

301.  Formula. — Principal  =  interest  ( rate  %  X  time). 

302.  Rules. — To  find  the  principal  when  the  interest,  rate 
per  cent,  and  time  are  given,  1.  Divide  the  given  interest  by 
the  interest  of  $1  for  the  given  time  and  rate  per  cent ,  and 
multiply  $1  by  the  quotient.  Or, 

2.  Divide  the  given  interest  by  the  decimal  corresponding  to 
the  interest  of  $1  for  the  given  time  and  rate  per  cent. 


Problem  V. 


Amount,  Hate  Her  Cent,  and.  Time  given,  to  find 

the  Principal. 


34.  What  principal,  on  interest  at  8%,  for  1  yr.  6  mo. 
18  da.,  will  give  an  amount  of  $730.60? 


1st  Process. 

$1,124  =  amount  of  $1. 

$1.124 )  $730,600  (  650. 
6744 

5620 

5620 


0 


The  interest  of  $1  for  1  yr.  6  mo. 
18  da.  is  $.124,  and  the  amount  is 
$1,124.  If  the  amount  of  $1  is 
$1,124,  it  will  take  as  many  times 
$1  to  yield  an  amount  of  $730.60 
as  $1,124  is  contained  times  in 
$730.60,  which  is  650.  650  times 
$1  is  $650,  the  required  principal. 
Or, 


190 


COMPLETE  ARITHMETIC. 


2d  Process. 

1.124  )  $730,600  (  $650. 
6744 

5620 

5620 


Since  the  amount  of  $1  is  1.124 
of  itself,  $730.60  is  1.124  of  the  re¬ 
quired  principal.  $730.60  -f- 1.124 
=  $650. 


35.  What  principal  on  interest,  at  5  %,  for  1  yr.  10  mo. 
12  da.,  will  amount  to  $70.24? 

36.  What  sum  of  money  put  at  interest, -at  7  %,  for  8 
mo.  18  da.,  will  amount  to  $567.09? 

37.  What  sum  of  money  put  at  interest,  at  8  %,  for  2 
yr.  1  mo.  15  da.,  will  amount  to  $421.20? 

38.  What  sum  of  money  put  at  interest  March  15,  1870, 
at  6  %,  will  amount  to  $2600.40,  Aug.  6,  1871? 

FORMULA  AND  RULE. 

303.  Formula. — Principal  =  amt.  -4-  [1  -f-  ( rate  %  X  time)"]. 

304.  Rules. — 1.  To  find  the  principal  when  the  amount, 
rate  per  cent,  and  time  are  given,  1.  Divide  the  given  amount 
by  the  amount  of  $1  for  the  given  time  and  rate  per  cent ,  and 
multiply  $1  by  the  quotient.  Or, 

2.  Divide  the  given  amount  by  the  decimal  corresponding  to 
the  amount  of  $1  for  the  given  time  and  rate  per  cent 

Note. — The  principal  thus  found  is  the  •present  worth  of  the  amount. 
See  Art.  306. 

REVIEW  OF  THE  FIVE  PROBLEMS. 

305.  The  formulas  for  the  five  problems  in  interest  are 
here  presented  together : 

Formulas. — 1.  Interest = principal  X  rate  %  X  time. 

2.  Rate  =  int  -4-  ( principal  X  1  %  X  time). 

3.  Time  =  interest  -4-  ( principal  X  rate  %). 

4.  Principal  =  interest  -4-  ( rate  %  X  time). 

5.  Principal  —  amt.  —5—  [1  — |-  ( fate%  X  time)]. 


PROBLEMS  IN  INTEREST. 


191 


WRITTEN  PROBLEMS. 

1.  What  is  the  interest  of  $205  for  2  yr.  5  mo.  24  da., 
at  7  %  ? 

2.  What  is  the  amount  of  $160,  from  Jan.  12,  1869,  to 
July  3,  1870,  at  8  %  ? 

*3.  At  what  rate  per  cent  will  $512.60  yield  $25,715 
interest  in  8  mo.  18  da.  ? 

*4.  The  principal  is  $126.75,  the  interest  $20,956,  and 
the  time  2  yr.  24  da.  :  what  is  the  rate? 

5.  How  long  will  it  take  $5000  to  produce  $1125  inter¬ 
est,  at  8  %  ? 

6.  The  principal  is  $326.50,  the  interest  $2.76,  and  the 
rate  8  %  :  what  is  the  time  ? 

7.  The  amount  is  $1563.75,  the  interest  $63.75,  and  the 
rate  7-J  %  :  what  is  the  time? 

8.  What  principal  will  yield  $1.36  interest  in  20  days, 
at  6%? 

9.  The  interest  on  a  certain  principal  from  Nov.  11, 
1857,  to  Dec.  15,  1859,  at  6%,  was  $4,474:  what  was  the 
principal  ? 

10.  What  principal,  at  7  %,  will  amount  to  $659.40  in  8 
months  ? 

11.  The  interest  is  $12.78,  the  time  1  yr.  2  mo.  6  da., 
and  the  rate  6  %  •  what  is  the  amount  ? 

12.  What  principal  will  amount  to  $609.20  in  4  mo.  18 
da.,  at  4  %  ? 

13.  What  principal  will  amount  to  $288.85  in  1  yr.  6 
mo. ,  at  6  %  ? 

14.  What  principal  will  produce  $21,757  interest,  from 
Jan.  1.  to  Oct.  20,  1869,  at  7  %  ? 

15.  How  long  will  it  take  any  principal  to  double  itself 
at  6  %  ?  At  4  %  ?  At  10  %  ? 

16.  What  is  the  amount  of  $420,  from  June  10,  1869, 
to  Jan.  21,  1870,  at  10  %? 

17.  What  sum,  bearing  interest  at  7  %,  will  yield  an  an¬ 
nual  income  of  $1000? 

*  Revised. 


192 


COMPLETE  ARITHMETIC. 


PRESENT  WORTH  AND  DISCOUNT. 

306.  The  most  common  application  of  Problem  Y  is  in 
computing  Present  Worth  and  Discount. 

307.  The  Present  Worth  of  a  debt  due  at  a  future 
time,  without  interest,  is  the  sum  or  principal  which,  at  the 
current  rate  of  interest,  will  amount  to  the  debt  when  it 
becomes  due. 

308.  Discount  is  the  amount  deducted  from  a  debt 
for  its  payment  before  it  is  due. 

309.  True  Discount  is  the  difference  between  a  debt, 
not  bearing  interest,  and  its  present  worth. 

True  discount  is  the  interest  on  the  present  worth  of  a  debt,  while 
simple  interest  is  computed  on  the  debt  itself.  The  difference  is  the 
interest  on  the  true  discount  for  the  time. 


WHITTEN  PROBLEMS. 


1.  What  is  the  present  worth  of  a  note  of  $212,  due  1 
year  hence,  without  interest,  the  current  rate  of  interest 
being  6  %  ?  What  is  the  true  discount  ? 


Process. 

$1.06  )  $212.00  (  200 
212 

$1  X  200  =  $200,  Present  worth. 
$212  —  200  =  $12,  True  discount. 


The  amount  of  $1  for  1  year, 
at  6%,  is  $1.06,  and  hence  the 
present  worth  of  $1.06,  due  1  year 
hence,  is  $1.  If  the  present  worth 
of  $1.06  is  $1,  the  present  worth 
of  $212.  is  as  many  times  $1  as 


$1.06  is  contained  times  in  $212,  which  is  200.  Or,  since  $1.06  is 
1.06  of  $1,  $212  is  1.06  of  its  present  worth.  $212  -4-  1.06  =$200. 


Note. — This  is  only  an  application  of  Problem  V.,  the  debt  being 
the  amount ,  the  present  worth  the  principal ,  and  the  true  discount  the 
interest. 


2.  What  is  the  present  worth  of  a  bill  of  $260  due  in  8 
months,  without  interest,  the  current  rate  of  interest  being 
6  %  ?  What  is  the  true  discount  ? 


PRESENT  WORTH  AND  DISCOUNT. 


193 


Find  the  present  worth  and  the  true  discount  of 

3.  $220  due  in  1  yr.  6  mo.,  without  interest,  current 
rate  7  %. 

4.  $145.60  due  in  8  mo.  12  da.,  without  interest,  cur¬ 
rent  rate  8  % . 

5.  $305.75  due  in  9  mo.  6  da.,  without  interest,  current 
rate  9  %. 

6.  $1250  due  in  1  yr.  7  mo.  21  da.,  without  interest, 
current  rate  5  % . 

7.  $1508  due  .in  90  days,  without  interest,  at  7  %. 

8.  $2040.50  due  in  36  days,  without  interest,  at  10  °f0. 

9.  $884,125  due  in  1  yr.  2  mo.  10  da.,  without  inter¬ 
est,  at  6  % . 

10.  What  is  the  difference  between  the  true  discount  of 
$216,  due  in  2  years,  without  interest,  and  the  interest  of  $216 
for  2  years,  at  8  %  ? 

What  is  the  difference  between  the  true  discount,  prin¬ 
cipal  due  in  given  time  without  interest,  and  the  interest  of, 

11.  $199.80  for  1  yr.  10  mo.,  at  6%?  At  12  %  ? 

12.  $666.40  for  2  yr.  4  mo.  15  da.,  at  6%?  At  8  %  ? 

13.  $534  for  1  yr.  1  mo.  18  da.,  at  6  %  ?  At  5  %  ? 

14.  $175.20  for  1  yr.  10  mo.  12  da.,  at  9  %  ? 

15.  $1250.60  for  1  yr.  7  mo.  24  da.,  at  5  %  ? 

16.  $884.12  for  96  days,  at  7J  %  ?  At  10  %  ? 

17.  How  large  a  note,  due  in  1  yr.  6  mo.,  with  interest 
at  7  %,  wrill  cancel  a  debt  of  $442,  due  in  1  yr.  6  mo.,  with¬ 
out  interest? 

18.  What  is  the  difference  in  the  present  value  of  a  cash 
payment  of  $345,  and  a  note  of  $371  due  in  9  months,  with¬ 
out  interest,  the  use  of  money  being  worth  8  %  ? 

FORMULAS  AND  RULES. 

310.  Formulas. — 1.  Present  worth  =  debt~  [i  +  (  rate  % 

X  time)']. 

2.  True  discount  —  debt  —  present  worth . 


C.Ar.— 17. 


194 


COMPLETE  ARITHMETIC. 


311.  Kules. — 1.  To  find  the  present  worth  of  a  debt  due 
at  a  future  time,  without  interest,  1.  Divide  the  debt  by  the 
amount  of  $1  for  the  given  time,  at  the  current  rate  of  interest, 
and  multiply  $1  by  the  quotient.  Or, 

2.  Divide  the  debt  by  the  decimal  corresponding  to  the  amount 
of  $1  for  the  given  time,  at  the  current  rate  of  interest. 


BANK  DISCOUNT. 


312.  Daflh  Discount  is  the  interest  on  a  note  for 


the  number  of  days  from 
the  time  it  is  discounted  to 
the  time  it  is  legally  due. 

313.  The  Proceeds 

of  a  note  are  its  face,  or 
the  sum  discounted,  less 
the  discount.  The  pro¬ 
ceeds  are  also  called  Avails 
and  Cash  Value. 

314.  Days  of  Grace 

are  the  three  days  allowed 
for  the  payment  of  a  note 
after  the  specified  time 
has  expired. 


Note. — A  note  is  payable,  or  nominally  due,  at  the  expiration  of  the 
specified  time,  but  it  does  not  mature,  or  become  legally  due,  until  the 
last  day  of  grace,  or  the  day  preceding,  when  the  last  day  of  grace 
falls  on  Sunday  or  a  legal  holiday.  The  date  of  expiration  and  the 
date  of  maturity  are  usually  written  with  a  line  between  them ;  thus, 
Jan.  9/  12. 


315.  When  a  bank  loans  money,  the  borrower  gives  his 
note  payable  at  a  specified  time,  without  interest.  This  note 
is  then  discounted  by  the  bank  for  the  actual  number  of 
days  in  the  time  plus  the  three  days  of  grace,  and  the  pro¬ 
ceeds  are  paid  to  the  borrower. 


316.  When  a  note  drawing  interest  is  discounted  by  a 


BANK  DISCOUNT. 


195 


bank,  the  discount  is  computed  on  the  amount  of  the  note 
at  the  time  of  its  maturity. 

Business  men  generally  discount  notes  and  bills,  not  drawing  in¬ 
terest,  by  deducting  the  interest  for  the  time,  with  or  without  grace 
as  per  agreement.  This  is  sometimes  called  Business  Discount.  The 
rate  of  interest  allowed  is  usually  greater  than  the  current  rate. 

Bills  due  in  three,  four,  or  six  months  are  often  discounted  by  de¬ 
ducting  5%  or  more  of  their  face,  without  regard  to  time.  This  is 
called  Per  Cent  Off  or  Trade  Discount. 

WRITTEN  PROBLEMS. 

1.  What  is  the  bank  discount  of  a  note  of  $350,  payable 
in  60  days,  discounted  at  10  %  ?  What  are  the  proceeds  ? 

Process. 

$350. 

.0105 

1750  Time  =60  da.  -f-  3  da.  =  63  da. 

3  50 

6  )  $3.6750 

$.6125  X  10  =  $6,125,  Bank  discount. 

$350  —  $6,125  =  $343,875,  Proceeds. 

2.  What  is  the  bank  discount  of  a  note  of  $250,  payable 
in  90  days,  discounted  at  8  %  ?  What  are  the  proceeds  ? 

Find  the  bank  discount  and  the  proceeds  of  a  note  of 

3.  $145,  payable  in  60  days,  discounted  at  7  %. 

4.  $80.50,  payable  in  30  days,  discounted  at  8  %. 

5.  $1000,  payable  in  90  days,  discounted  at  1\%. 

6.  $750,  payable  in  45  days,  discounted  at  9  %. 

7.  $1250,  payable  in  100  days,  discounted  at  6%. 

8.  $56,  dated  Jan.  1,  1870,  payable  May  1,  1870,  dis¬ 
count  6  %. 

9.  $120,  dated  Apr.  3,  1869,  payable  June  15,  1869,  dis¬ 
count  8  %. 

10.  $500,  dated  Dec.  15,  1870,  payable  Feb.  18,  1871, 
discount  9  %. 

11.  $8.75,  dated  Nov.  21,  1870,  payable  Mch.  12,  1871, 
discount  9  %. 


196 


COMPLETE  ARITHMETIC. 


12.  $400,  dated  Mch.  4,  1871,  payable  July  24,  1871, 
discount  5  %  ? 

13.  What  is  the  difference  between  the  bank  discount 
and  the  true  discount  of  $1319.50,  due  in  90  days,  dis¬ 
counted  at  6  °/0  ?  [For  meaning  of  “due,”  see  Note,  p.  194.] 

14.  What  is  the  difference  between  the  bank  discount 
and  the  true  discount  of  $768,  due  in  108  days,  discount 
8%? 

15.  What  is  the  difference  between  the  bank  discount 
and  the  true  discount  of  $3330.80,  due  in  45  days,  dis¬ 
count  7  %  ? 

16.  A  note,  dated  Apr.  10,  1871,  is  payable  in  90  days: 
what  is  the  time  of  its  maturity? 

Note. — The  date  of  maturity  is  found  by  counting  forward  the 
number  of  days  plus  three  days,  when  the  time  is  expressed  in  days ; 
and  the  number  of  calendar  months  plus  three  days,  when  the  time 
is  expressed  in  months. 

17.  A  note,  dated  Feb.  6,  1868,  was  payable  in  60  days : 
what  was  the  date  of  its  maturity  ? 

18.  A  note,  dated  Aug.  9,  1870,  is  payable  4  months 
from  date :  what  is  the  date  of  its  maturity  ? 

19.  A  note  of  $460,  dated  Apr.  3,  1870,  and  payable  in 
<90  days,  with  interest  at  6%,  was  discounted  May  10,  1870, 
at  8  %  :  what  were  the  proceeds  ? 

/ 

Suggestion.— Find  the  amount  of  $460  for  93  days,  at  6%,  and 
then  discount  this  amount  for  56  days,  at  8%. 

20.  A  note  of  $125,  dated  May  21,  1870,  and  payable  in 
60  days,  wTith  interest  at  6  %,  was  discounted  May  25,  1870, 
at  10  %  :  what  were  the  proceeds  ? 

21.  A  note  of  $1000,  dated  Aug.  15,  1869,  and  payable 
in  6  months,  with  interest  at  7  %,  was  discounted  Nov.  27, 
1869,  at  9  % :  what  was  the  bank  discount  ? 

Suggestion. — Compute  the  interest  for  6  mo.  3  da.,  and  the  dis¬ 
count  for  83  days. 

22.  A  note  of  $90,  dated  Apr.  12,  1870,  and  payable  in 


BANK  DISCOUNT. 


197 


4  months,  with  interest  at  5  %,  was  discounted  June  1, 
1870,  at  7  %  :  what  were  the  proceeds? 

23.  A  note  of  $650,  dated  Mch.  2,  1869,  payable  Apr. 
1,  1870,  and  indorsed  $300  Oct.  1,  1869,  was  discounted 
Feb.  3,  1870:  what  were  the  proceeds  at  6  %  ? 

24.  For  what  sum  must  a  note,  payable  in  60  days,  be 
drawn  to  produce  $493,  when  discounted  at  8  %  ? 


Process. 

$1  —  $.014  =  $.986,  Proceeds  of  $1. 
$493  -f-  $.986  =  500 
$1  X  500  =  $500,  Face  of  note. 


Since  the  proceeds  of  $1  are 
$.986,  it  will  require  as  many 
times  $1  to  produce  $493  as 
$.986  is  contained  times  in  $493, 
which  is  500 ;  and  500  times  $1 
=  $500. 


25.  For  what  sum  must  a  note,  payable  in  90  days,  be 
drawn  to  produce  $1969,  when  discounted  at  6%? 

26.  What  must  be  the  face  of  a  note,  dated  July  5,  1871, 
and  payable  in  4  months,  to  produce  $811,  when  discounted 
at  9  %  ? 

27.  What  must  be  the  face  of  a  note,  dated  Jan.  10, 
1870,  and  payable  in  3  months,  to  produce  $1938,  when 
discounted  at  12  %  ? 

28.  A  merchant  discounted  a  bill  of  $750,  payable  in  4 
months,  by  deducting  the  interest  for  the  time  without 
grace,  at  10  % '  what  were  the  cash  proceeds  of  the  bill  ? 

29.  A  note  of  $340,  due  in  9  months,  without  interest, 
was  discounted  by  deducting  the  interest  for  the  time,  at 
8  % :  what  was  the  cash  value  of  the  note  ? 

30.  A  merchant  having  sold  a  bill  of  goods  amounting 
to  $1030,  on  three  months’  time,  allowed  5  %  off  for  cash : 
what  were  the  cash  proceeds  of  the  sale? 

31.  A  retail  dealer  having  bought  $950  worth  of  goods, 
on  6  months’  time,  cashed  the  bill  for  off:  what  were 
the  cash  proceeds? 

32.  A  merchant  bought,  March  20,  1870,  a  bill  of  goods 
amounting  to  $3540,  on  three  months’  time,  but,  being 
offered  5  %  off  for  cash,  he  borrowed  the  money  at  a  bank 


198 


COMPLETE  ARITHMETIC. 


for  the  time,  at  10  </c,  and  cashed  the  bill.  How  much  did 
he  gain  by  the  transaction? 

FORMULAS  AND  RULES. 

317.  Formulas. — 1.  Bank  discount  =  sum  discounted  X 

(int.  of  $1  for  the  days  -j-  3  days'). 

2.  Business  discount = sum  discounted  X 
int.  of  SI  for  the  time. 

3.  Proceeds —sum  discounted — discount. 

318.  Rules. — 1.  To  compute  bank  discount,  Find  the  in¬ 
terest  on  the  sum  discounted  at  the  given  rate  per  cent  and  for 
the  actual  number  of  days  in  the  time  plus  three  days. 

2.  To  compute  business  discount,  Find  the  interest  on  the 
sum  discounted ,  at  the  given  rate  per  cent  and  for  the  given 
time. 

3.  To  find  the  proceeds,  Subtract  the  discount  from  the  sum 
discounted. 

4.  To  find  the  face  of  a  note  to  yield  given  proceeds, 
Divide  the  given  proceeds  by  $1  minus  the  interest  of  SI  for  the 
given  time  with  grace ,  and  multiply  $1  by  the  quotient. 

Notes. — 1.  In  discounting  a  note  bearing  interest,  the  interest  i9 
computed  by  months  or  by  days,  according  as  the  time  is  expressed 
in  the  note,  but  the  discount  is  usually  computed  by  days. 

2.  Business  discount  is  computed  by  months  or  days,  according  as 
the  time  is  expressed  in  the  paper  discounted,  and  with  or  without 
grace.  In  these  respects  it  differs  from  bank  discount. 

3.  Bank  discount  is  not  only  interest  paid  in  advance,  but  the  in¬ 
terest  is  computed  on  both  the  proceeds  and  the  discount.  The  bor¬ 
rower  pays  interest  on  more  money  than  he  receives. 


NOTES,  DRAFTS,  AND  BONDS. 

I.  PROMISSORY  NOTES. 

319.  A  Promissory  )Note  is  a  written  agreement  by 
one  party  to  pay  to  another  a  specified  sum  of  money  at  a 
specified  time.  The  sum  specified  in  the  note  is  called  its 
Face. 

The  person  who  signs  a  note  is  called  its  Maker ;  the  person  to 
whom  it  is  payable  is  the  Payee;  and  its  owner  is  the  Holder. 


PROMISSORY  NOTES. 


199 


320.  A  Joint  Note  is  a  note  signed  by  two  or  more 
persons  who  are  jointly  liable  for  its  jmyment. 

321.  A  Joint  and  Several  Note  is  a  note  signed 
by  two  or  more  persons  who  are  both  jointly  and  singly 
liable  for  its  payment. 

322.  An  Indorser  is  a  person  who  signs  his  name  on 
the  back  of  a  note  as  security  for  its  payment. 

323.  The  following  are  the  more  common  forms  of  pro¬ 
missory  notes : 


FORM  I. —  Demand  Note. 

$95XW  Nashville,  Tenn.,  May  1,  1870. 

For  value  received ,  I  promise  to  pay  to  John  Wilson ,  on 
demand ,  Ninety-five  Dollars. 

Henry  Smith. 


FORM  II. —  Time  Note. 

$95t5_o^  St.  Louis,  Mo.,  May  1,  1870. 

Ninety  days  after  date ,  I  promise  to  pay  to  John  Wilson, 
or  hearer,  Ninety-five  Dollars,  with  interest,  for  value  re¬ 
ceived. 

Henry  Smith. 


FORM  III. —  Joint  and  Several  Note. 


$S5y$y.  Louisville,  Ky.,  March  12,  1870. 

Four  months  after  date,  ice  jointly  and  severally  promise 
to  pay  Henry  Cooke,  or  order,  Ninety-five  Dollars,  with  in¬ 
terest  at  8  % ,  for  value  received. 

Thomas  Hughes, 

i 

Charles  G.  Knight. 


200 


COMPLETE  ARITHMETIC. 


FORM  IV. — Note  Payable  at  a  Bank. 

$500.  Baltimore,  Md.,  Apr.  10,  1870. 

Sixty  days  after  date ,  we  promise  to  pay  to  Wilson ,  Hinkle 
&  Co .,  or  order ,  at  the  First  National  Bank,  Five  Hundred 
Dollars ,  for  value  received.  Charles  Cooke  &  Co. 

Remarks. — 1.  A  note  should  contain  the  words  “  value  received,” 
otherwise  the  holder  may  be  required  to  prove  that  the  maker  re¬ 
ceived  its  value. 

2.  When  the  time  for  the  payment  of  a  note  is  not  specified,  it  is 
due  on  demand.  If  the  place  of  payment  is  not  mentioned,  it  is  pay¬ 
able  at  the  maker’s  residence  or  place  of  business. 

3.  When  a  note  contains  the  words  “with  interest,”  and  no  rate  is 
specified  (Form  II.),  interest  accrues  at  the  legal  rate.  If  the  words 
“  with  interest,”  are  omitted,  no  interest  accrues  until  after  maturity, 
when  the  note  draws  interest,  at  the  legal  rate,  until  paid. 

324.  A  Negotiable  Note  is  one  which  may  be  bought 
and  sold. 

A  note  is  negotiable  when  it  is  made  payable  “  to  the  bearer,”  or 
to  the  payee  “or  bearer,”  or  to  the  payee  “or  order,”  or  “to  the 
order  of”  the  payee.  A  note  drawn  as  in  Form  I.  is  not  negotiable. 

A  note  made  payable  to  the  bearer  is  negotiable  without  indorse¬ 
ment.  U.  S.  treasury  notes  and  bank  notes,  used  as  money,  are  pay¬ 
able  to  the  bearer,  and  are  transferred  without  indorsement. 

A  note  payable  to  order  must  be  indorsed  by  the  payee  before  it  is 
negotiable.  When  the  payee  indorses  a  note  by  simply  writing  his 
name  on  the  back,  it  is  called  an  indoi'sement  in  blank ,  and  the  note 
is  payable  to  the  holder.  When  the  indorser  orders  the  payment 
to  be  made  to  a  particular  person,  as:  “Pay  to  Charles  Williams,” 
it  is  called  a  special  indorsement. 

325.  If  the  maker  of  a  note  fails  to  pay  it  at  maturity, 
a  written  notice  of  the  fact,  made  by  a  notary  public,  is 
served  on  the  indorsers,  who  are  responsible  for  the  payment 
of  the  note.  Such  a  notice  is  called  a  Protest. 

Note. — A  protest  must  be  made  on  the  day  a  note  matures,  and 
it  must  be  sent  on  that  day  or  the  next,  otherwise  the  indorsers 
are  not  responsible. 


BILLS  OF  EXCHANGE. 


201 


IL  DRAFTS,  OR  BILLS  OF  EXCHANGE. 

326.  A  Draft  is  an  order  made  by  one  person  upon 
another  to  pay  a  specified  sum  to  a  third  person  named.  It 
is  also  called  a  Bill  of  Exchange. 

The  person  who  makes  the  order  is  called  the  Drawer ;  the  per¬ 
son  to  whom  it  is  addressed  is  called  the  Drawee;  and  the  person 
to  whom  the  money  is  payable  is  the  Payee. 

327.  The  following  are  the  common  forms  of  domestic 
drafts : 


FORM  I. — Sight  Draft. 

$100.  Cincinnati,  O.,  Oct.  1,  1870. 

Pay  to  the  order  of  Bartlit  &  Smith ,  One  Hundred 
Dollars,  and  place  to  the  account  of 

Charles  S.  Kelley. 

To  George  Brown,  Esq.,  New  York. 


FORM  II. — Time  Draft. 

$100.  Cincinnati,  O.,  Oct.  1,  1870. 

Thirty  days  after  sight  [or  date ],  pay  to  the  order  of 
Bartlit  &  Smith,  One  Hundred  Dollars ,  and  place  to  the 
account  of 

Charles  S.  Kelley. 

To  George  Brown,  Esq.,  New  York. 


When  the  drawee  accepts  a  draft,  he  writes  the  word  “Accepted,'’ 
with  the  date,  across  the  face,  and  signs  his  name,  thus :  “  Accepted, 
Oct.  3,  1870 — George  Brown.”  The  draft  is  then  called  an  Accept¬ 
ance,  and  the  acceptor  is  responsible  for  its  payment. 

A  draft  made  payable  to  bearer  or  order  is  negotiable,  like  a 
promissory  note,  and  is  subject  to  protest  in  case  payment  or  accept¬ 
ance  is  refused. 


202 


COMPLETE  ARITHMETIC. 


Notes. — 1.  In  most  of  the  states  both  time  and  sight  drafts  are 
entitled  to  three  days  of  grace.  In  New  York  no  grace  is  allowed 
©n  sight  drafts. 

2.  When  a  draft  is  drawn,  “acceptance  waived,”  it  is  not  subject  to 
protest  until  maturity  ;  and  when  an  indorser  writes  over  his  name, 
“  demand  and  notice  waived,”  a  protest  in  his  case  is  not  necessary. 

3.  The  liability  of  an  indorser  of  a  note  or  draft  may  be  avoided 
by  his  writing  over  his  indorsement,  “  without  recourse.” 

328.  A  Domestic  or  Inland  Bill  is  a  draft  which 
is  payable  in  the  country  where  it  is  drawn. 

A  Foreign  Bill  is  a  draft  which  is  drawn  in  one 
country  and  is  payable  in  another. 

329.  Fxeliange  is  the  process  of  making  payments  at 
distant  places  by  the  remittance  of  drafts,  instead  of  money. 

When  a  draft  can  be  bought  for  its  face,  it  is  said  to  be  at  par ; 
when  the  cost  is  less  than  the  face,  it  is  below  par ,  or  at  a  discount ; 
and  when  the  cost  is  more  than  the  face,  it  is  above  par ,  or  at  a  pre¬ 
mium.  The  rate  per  cent,  which  the  cost  of  a  draft  is  more  or  less 
than  its  face,  is  called  the  Rate  of  Exchange. 

Notes. — 1.  The  rate  of  exchange  between  two  places  depends  chiefly 
on  their  relative  trade.  If  Cincinnati  owes  New  York,  drafts  on 
New  York  are  at  a  premium  in  Cincinnati;  if  New  York  owes  Cin¬ 
cinnati,  drafts  on  New  York  are  at  a  discount;  if  the  trade  of  the 
two  cities  with  each  other  is  equal,  exchange  is  at  par. 

2.  In  foreign  exchange,  drafts  are  expressed  in  the  currency  of 
the  country  on  which  they  are  drawn.  The  comparative  value  of 
the  money  of  two  countries  is  called  the  Par  of  Exchange. 


WRITTEN  PROBLEMS. 

1.  What  is  the  cost  of  a  draft  on  New  York  for  $800, 
exchange  being  f  %  premium? 

Process  :  $800  X  -00f  =  $6,  Bern.  $800  +  $6  =  $806,  Cost. 

2.  What  is  the  cost  of  a  draft  on  New  Orleans  for  $1250, 
at  ^  %  discount? 

3.  What  is  the  cost  of  a  draft  on  Philadelphia  for  $1050, 
at  \  %  premium  ? 

4.  A  merchant  in  St.  Louis  wishes  to  remit  $2500  bv 

«/ 

draft  to  New  York:  what  will  be  the  cost  of  the  draft,  ex¬ 
change  being  1J  %  premium? 


EXCHANGE. 


203 


5.  What  will  be  the  cost  of  a  draft  for  $500,  payable 
in  30  days  after  sight,  exchange  being  1  %  premium,  and 
interest  6  %  ? 

Process. 

$500. 

$500  X  .0055  =  $2.75  Discount  at  6%  for  S3  days. 

$497.25  Proceeds  of  Draft  (cost  at  par). 

$500  X  *01  =  5.00  Premium  atlfc. 

$502.25  Cost  of  Draft.  " 

Note. — If  preferred,  the  face  may  be  multiplied  by  the  cost  of 
$1,  which  is  $1  —  $.0055  +  $.01,  or  $1.0045.  $500  X  1-0045  = 

$502.25. 

6.  What  will  be  the  cost  of  a  draft  for  $650,  payable  in 
60  days  after  sight,  exchange  being  f  %  premium,  and  in¬ 
terest  8  %  ? 

7.  What  will  be  the  cost  of  a  draft  for  $320,  payable  in 
45  days  after  sight,  exchange  being  f  %  discount,  and  in¬ 
terest  7  %  ? 

8.  How  large  a  sight  draft  can  be  bought  for  $259.52, 
exchange  being  If  %  premium  ? 

Process:  $259.52 -f-  1.01  $  =  $250,  Face  of  Draft.  (Case  IV.) 

9.  How  large  a  sight  draft  can  be  bought  for  $962.85, 
exchange  being  If  %  discount? 

10.  A  sight  draft,  bought  at  f  %  premium,  cost  $1256.25  : 
what  was  its  face? 

11.  How  large  a  draft,  payable  30  days  after  sight,  can 
be  bought  for  $502.25,  exchange  being  1  and  interest  6  %  ? 

Process. 

$1  —  $.0055  =  $.9945  Proceeds  of  $1  discounted  for  33  days. 

$.9945  +  $.01  =  $1.0045  Cost  of  $1. 

$502.25  -s-  $1.0045  =  $500  Face  of  Draft. 

12.  How  large  a  draft,  payable  60  days  after  sight,  can 
be  bought  for  $798.80,  exchange  being  If  %  premium,  and 
interest  8  %  ? 

13.  A  draft,  payable  in  30  days  after  sight,  was  bought 
for  $352.62,  exchange  being  If  %  discount,  and  interest 
6  %  •  what  was  its  face? 


204 


COMPLETE  ARITHMETIC. 


III.  BONDS. 

330.  The  interest  bearing  notes  issued  by  nations,  states, 
cities,  railroad  companies,  and  other  corporations,  as  a 
means  of  borrowing  money,  are  called  Bonds. 

Bonds  are  issued  under  seal  in  denominations  of  convenient  size, 
with  interest  usually  payable  annually  or  semi-annually,  and  they 
are  made  negotiable  like  certificates  of  capital  stock.  (Art.  243.) 

The  Coupons  attached  to  transferable  bonds,  are  due-bills  for  the 
interest,  which,  as  the  interest  becomes  due,  are  cut  off  and  pre¬ 
sented  for  payment. 

331.  The  several  classes  of  bonds  issued  by  the  United 
States  Government  are  called  United  States  Securities ,  or 
Government  Securities. 

[Note. — When  the  first  edition  of  this  treatise  was  published  (in 
1870),  the  principal  United  States  ♦Securities  were  known  as  the 
“Five-Twenties,”  the  “Ten-Forties,”  and  the  “  Sixes  of  1881.”  The 
5-20’s  and  10-40’s,  and  most  of  the  6’s  of  ’81  were  redeemed  prior  to 
1880,  and  new  series  of  bonds,  at  lower  rates  of  interest,  have  since 
been  issued.  From  1862  to  1878  inclusive,  the  value  of  gold  coin 
was  greater  than  that  of  paper  money,  and  gold  was  quoted  at  a 
premium.  In  the  problems  in  this  edition,  the  descriptions  of  the 
different  classes  of  U.  S.  bonds  are  omitted,  but  the  gold  quotations 
are  retained,  since  they  illustrate  an  important  principle  of  per¬ 
centage.] 

332.  The  market  value  of  United  States  bonds  is  quoted 
at  a  certain  per  cent  of  their  par  value  or  face.  Bonds 
quoted  at  110  are  worth  110%  of  their  face;  that  is,  are 
10%  above  par.  The  quotation  includes  accrued  interest. 

WHITTEN  PROBLEMS. 

1.  When  U.  S.  bonds  are  quoted  at  109J,  what  will 
three  $500  bonds  cost? 

2.  When  U.  S.  bonds  are  worth  114,  what  will  $1250  in 
bonds  cost? 

3.  A  widow  invested  $4725  in  U.  S.  bonds  at  105 :  what 
amount  in  bonds  did  she  receive? 

4.  A  broker  invested  $26250  in  bonds  at  106^,  and  sold 
them  at  109:  how  much  did  he  gain? 


ANNUAL  INTEREST. 


205 


5.  When  gold  is  at  115,  what  amount  in  currency  can 
be  bought  for  $8500  in  gold? 

6.  At  112,  what  amount  in  gold  can  be  bought  for 
$1400  in  treasury  notes? 

7.  When  gold  was  at  115,  and  U.  S.  bonds  at  105,  what 
was  a  $500  bond  worth  in  gold? 

8.  When  gold  was  at  120,  and  U.  S.  bonds  at  115, 
what  was  the  gold  value  of  a  $1000  bond? 

9.  When  gold  was  at  110,  and  U.  S.  bonds  at  108, 
what  was  the  gold  value  of  a  $500  bond  ? 

10.  When  gold  was  worth  112J,  what  was  the  gold  value 
of  a  dollar  treasury  note? 

11.  When  gold  was  at  115,  what  was  the  semi-annual 
interest  in  currency  on  $9500  in  U.  S.  10^  bonds,  gold? 

12.  When  gold  was  at  120,  what  rate  per  cent  in  currency 
was  the  interest  on  U.  S.  bonds  bearing  W/0  interest,  gold. 

*  13.  A  college  invested  its  endowment  fund  in  U.  S.  5^ 
bonds  (gold) :  what  rate  of  interest  in  currency  did  it  re¬ 
ceive  when  gold  was  at  120?  At  110?  At  par? 

ANNUAL  INTEREST. 

333.  When  a  note  reads  “  with  interest  payable  an¬ 
nually,”  the  interest  on  the  face  of  the  note,  due  at  the 
close  of  each  year,  is  called  Annual  Interest . 

334.  When  annual  interest  is  not  paid  at  the  close  of 
the  year,  when  due,  it  draws  simple  interest  until  paid. 

In  Ohio  and  several  other  states,  the  annual  interest,  if  not  paid 
when  due,  draws  simple  interest  at  the  legal  rate  (Art.  282),  whatever 
may  be  the  rate  of  interest  prescribed  for  the  principal. 

WRITTEN  PROBLEMS. 

1.  A  note  of  $500,  dated  May  10,  1870,  is  due  in  4  years, 
-with  interest  at  6^,  payable  annually:  if  both  interest 
and  principal  remain  unpaid,  what  will  be  the  amount 
due  on  the  note  at  maturity? 


206 


COMPLETE  ARITHMETIC. 


Process. 

$500  X  .06  =  $30,  Interest  on 'principal  due  annually. 

$30X4=  $120,  Total  annual  interest. 

$30  X  -06  X  3  =  5.40,  Simple  interest  on  1st  annual  interest  for  3  yrs. 

$30  X  .06  X  2  =  3.60,  “  “  “  2d  “  “  “  2  “ 

$30  X  -06  X  1  =  1-80,  “  “  “  3 d  “  “  “  1  “ 

$130.80,  Total  interest  due  at  maturity  of  note. 

500.00 

$630.80,  Amount  due  at  maturity  of  note. 

The  $30  annual  interest,  due  at  the  close  of  the  1st  year,  being 
unpaid,  draws  simple  interest  until  paid,  or  for  3  years;  the  $30 
annual  interest  due  at  the  close  of  the  2d  year,  draws  interest  for  2 
years ;  and  the  $30  annual  interest  due  at  the  close  of  the  3d  year, 
draws  interest  for  1  year.  The  fourth  annual  interest  is  paid  when 
due.  Hence,  the  total  interest  due  at  the  maturity  of  the  note,  con¬ 
sists  of  (1)  the  annual  interest  for  1  year  ($30)  multiplied  by  4,  the 
number  of  years;  and  (2)  the  simple  interest  on  the  $30  annual  in¬ 
terest  for  3  years,  2  years,  and  1  year,  or  for  6  years.  The  amount 
due  is  $500  +  $30  X  4  +  $1.80  X  6. 

2.  A  note  of  $750,  with  interest  payable  annually,  at  8%, 
was  paid  3  yr.  3  mo.  18  da.  after  date,  and  no  interest  had 
been  previously  paid :  what  was  the  amount  due  ? 

Process. 

$750  X  .08  =  $60.  Interest  on  face  due  annually. 

$60  X  3.3  =$198.00,  Total  annual  int.  for  3  yr.  3  mo.  18  da.  (3.3  yr.) 

$60  X  *312  =  $18.72,  Simple  interest  on  $60  for  3  yr.  10  mo.  24  da. 

$750. 

$966.72,  Amount  due. 

The  first  annual  interest  draws  simple  interest  for  2  yr.  3  mo. 
18  da. ;  the  second,  for  1  yr.  3  mo.  18  da. ;  and  the  third,  for  3  mo. 
18  da. ;  and  hence,  the  simple  interest  on  the  several  annual  interests 
equals  the  interest  of  $60  for  2  yr.  3  mo.  18  da.  + 1  yr.  3  mo.  18  da. 
+  3  mo.  18  da.,  or  for  3  yr.  10  mo.  24  da.  The  amount  due  consists 
of  (1)  the  principal ;  (2)  the  total  annual  interest;  and  (3)  the  simple 
interest  on  the  annual  interest. 

3.  A  note  of  $1000,  with  annual  interest  at  6%,  is  due 
4  yr.  6  mo.  after  date :  no  interest  having  been  paid,  what 
will  be  due  at  maturity? 


ANNUAL  INTEREST. 


207 


4.  A  man  bought  a  farm  for  $3500,  to  be  paid  in  4  years, 
with  annual  interest  at  6%,  but  failed  to  pay  the  interest: 
what  was  due  at  the  close  of  the  4th  year  ? 

5.  $650.  New  York,  July  1,  1869. 

On  the  first  day  of  January,  1872,  for  value  received,  I 
promise  to  pay  John  Black,  or  order,  six  hundred  and  fifty 

dollars,  with  interest,  payable  annually,  at  7  % . 

Charles  Church. 

If  no  interest  be  paid  on  the  above  note,  what  will  be 
due  at  maturity? 

6.  If  the  above  note  and  interest  be  not  paid  until  Sept. 
13,  1872,  what  will  be  the  amount  due  ? 

7.  A  note  of  $800,  dated  March  18,  1867,  and  due  in  3 
years,  with  interest  at  6%,  payable  annually,  has  the  fol¬ 
lowing  indorsements:  Oct.  24,  1868,  $150:  Nov.  12,  1869, 
$240.  What  was  the  amount  due  March  18,  1870? 


Process. 


$800  X  -06  =  $48, 
$48  X  2  =  $96, 
$48  X  -06  =  2.88, 
$98.88, 
800 

$898.88, 


Payment,  $150. 
Int.  on  same,  3.60. 


1  =$153.60, 
*  $745.28, 


First  annual  interest. 

Annual  interest  due  Mch.  18,  1869. 
Interest  on  1st  annual  interest. 
Interest  due  Mch.  18,  1869. 

Amount  due  Mch.  18,  1869. 

Amount  of  $150,  Mch.  18,  1869. 
New  principal. 


$745.28  X  .06  =  $44,716,  Interest  due  Mch.  18,  1870- 
$789,996,  Amount  due  Mch.  18,  1870. 


Payment,  $240. 
Int.  on  same,  5.04. 


=$245.04, 

$544,956, 


Amount  $240,  Mch.  18, 1870. 
Amount  due  at  maturity. 


Note. — The  annual  interest  and  the  interest  on  the  same  are  com¬ 
puted  to  the  close  of  the  year  in  which  the  first  payment  is  made,  and 
the  interest  on  the  payment  is  computed  to  the  same  date.  The  differ¬ 
ence  between  the  amount  of  the  face  of  the  note  and  the  amount  of  the 
payment  is  the  new  principal  for  the  third  year. 


8.  A  note  of  $500,  dated  Jan.  15,  1865,  and  due  in  2 
years,  with  interest  at  10%,  payable  annually,  is  indorsed 


208 


COMPLETE  ARITHMETIC. 


as  follows:  May  21,  1866,  $100;  Mch.  9,  1867,  $200.  What 
was  the  amount  due  July  15,  1867  ? 

335.  Rules. — 1.  To  compute  unpaid  annual  interest, 

Compute  the  interest  on  the  'principal  for  the  entire  time  it  is 
on  interest,  and  the  interest  on  each  year's  interest  for  the  time 
it  is  unpaid .  The  sum  of  the  principal ,  the  interest  on  the 
principal,  and  the  interest  on  the  unpaid  interest  will  be  the 
amount  due. 

Note. — Instead  of  computing  the  interest  on  the  several  annual 
interests  separately,  simple  interest  may  be  computed  on  one  year's  inter¬ 
est  for  a  time  equal  to  the  sum  of  the  periods  of  time  the  several  annual 
interests  are  unpaid.  For  rate  of  interest,  see  Art,  334. 

2.  To  compute  annual  interest  when  partial  payments 
have  been  made,  1.  Compute  the  interest  on  the  principal  to  the 
end  of  the  first  year  in  which  any  payment  is  made,  and  also 
the  interest  on  the  unpaid  annual  interest  to  the  same  date,  and 
form  the  amount. 

2.  Compute  the  interest  on  the  payment  or  payments  to  the 
end  of  the  year,  and  form  the  amount. 

3.  Subtract  the  amount  of  the  payment  or  payments  from  the 
amount  of  the  principal  and  interest,  and  taking  the  difference 
for  a  new  principal,  proceed  as  before  with  succeeding  pay¬ 
ments,  making  the  date  of  settlement  the  last  date. 

Note. — 1.  This  rule  is  based  on  the  principle  that  the  payments 
with  added  interest  should  be  applied  first  to  the  discharge  of  the 
accrued  interest  at  the  end  of  a  year,  and  then  to  the  discharge  of  the 
principal.  When  the  amount  of  the  payment  or  payments  will  not 
cancel  all  the  interest  due,  the  unpaid  interest  draws  simple  interest  to 
the  end  of  the  next  year  in  which  a  payment  is  made. 


COMPOUND  INTEREST. 

336.  Compound  Interest  is  interest  on  the  princi¬ 
pal  and  also  on  the  interest  which,  at  regular  intervals  of 
time,  is  added  to  the  principal.  It  is  generally  compounded 
annually,  semi-annually,  or  quarterly. 


COMPOUND  INTEREST. 


209 


WRITTEN  PROBLEMS. 


1.  A  note  of  $450  is  due  in  3  years  with  interest  at  6  % , 
compounded  annually.  What  will  be  the  amount  due  at 
maturity?  What  will  be  the  compound  interest  due? 


$450 

_ _ .06 

$27.00 

450. 

$477 

.06 

$28.62 
477. 
$505.62 
_ M 

$30.3372 

505.62 

$535.9572 

450. _ 

$85.9572 


Process. 

ls£  year’s  interest. 

2 d  principal. 

2d  year’s  interest. 

2>d  principal. 

3 d  year’s  interest. 

Amount  due  at  the  end  of  the  third  year. 
Compound  interest  at  the  end  of  the  third  year. 


2.  What  is  the  amount  of  $600  for  4  years  at  5  %,  com¬ 
pounded  annually  ?  What  is  the  compound  interest  ? 

3.  What  is  the  compound  interest  of  $1500  for  3  years 
at  10  %  ?  At  8  %  ? 

4.  What  is  the  amount  of  $800  for  2  years  at  6  %,  com¬ 
pounded  semi-annually  ? 

Suggestion. — Compute  the  interest  at  3  %. 

5.  What  is  the  compound  interest  of  $650  for  3  yr.  4  mo. 
12  d.  at  6  %  per  annum  ? 

337.  Rule. — To  compute  compound  interest,  Find  the 
amount  of  the  given  principal  for  one  interval  of  time;  then , 
taking  this  amount  as  a  new  principal,  find  the  amount  for  the 
second  interval,  and  so  continue  for  the  entire  time.  The  dif¬ 
ference  between  the  last  amount  and  the  principal  is  the  com¬ 
pound  interest  for  the  time. 

C.Ar.— 18. 


210 


COMPLETE  ARITHMETIC. 


Notes. — 1.  When  the  interest  is  compounded  semi-annually,  the 
rate  per  cent,  is  one  half  the  yearly  rate,  and  when  compounded 
quarterly,  it  is  one  fourth  the  yearly  rate. 

2.  When  the  time  contains  years,  months,  and  days,  the  amount 
is  found  for  the  number  of  whole  intervals  in  the  time,  and  then  the 
interest  is  computed  on  this  amount  for  the  remaining  months  and 
days. 

338.  Compound  interest  is  usually  computed  by  the  aid 
of  a  table  giving  the  amount  of  $1  at  several  different  rates 
per  cent,  and  for  any  number  of  years  which  may  be  in¬ 
cluded. 


339.  A  Table 

Showing  the  amount  of  $1  at  compound  interest,  at  3,  4,  5,  6, 
7,  or  8  per  cent.,  for  any  number  of  years  from  1  to  25. 


YUS. 

3  PER  CENT. 

4  PER  CENT. 

5  PER  CENT. 

6  PER  CENT. 

7  PER  CENT. 

8  PER  CENT. 

I 

1.03 

1.04 

1.05 

1.06 

1.07 

1.08 

o 

1 .0009 

1.0816 

1.1025 

1.1236 

1.1449 

1.1664 

3 

1.092727 

1.124864 

1.157625 

1.191016 

1.225043 

1.259712 

4 

1.125509 

1.169859 

1.215506 

1.262477 

1.310796 

1.360489 

5 

1.159274 

1.216653 

1.276282 

1.338226 

1.402552 

1.469328 

6 

1.194052 

1.265319 

1.340096 

1.418519 

1.500730 

1.586874 

7 

1.229874 

1.315932 

1.407100 

1.503630 

1.605781 

1.713824 

8 

1.266770 

1.368569 

1.477455 

1.593848 

1.718186 

1.850930 

9 

1.304773 

1.423312 

1.551328 

1.689479 

1.838459 

1.999005 

10 

1.343916 

1.480244 

1.628895 

1.790848 

1.967151 

2.158925 

11 

1.384234 

1.539454 

1.710339 

1.898299 

2.104852 

2.331639 

12 

1.425761 

1.601032 

1.795856 

2.012196 

2.252192 

2.518170 

13 

1.468534 

1.665074 

1.885649 

2.132928 

2.409845 

2.719624 

14 

1.512590 

1.731676 

1.979932 

2,260904 

2.578534 

2.937194 

15 

1.557967 

1.800944 

2.078928 

2.396558 

2.759032 

3.172169 

If) 

1.604706 

1.872981 

2.182875 

2.540352 

2.952164 

3.425943 

17 

1.652848 

1.947900 

2.292018 

2.692773 

3.158815 

3.700018 

18 

1.702433 

2.025817 

2.406619 

2.854339 

3.379932 

3.996019 

19 

1.753506 

2.106849 

2.526950 

3.025600 

3.616528 

4.315701 

20 

1.806111 

2.191123 

2.653298 

3.207135 

3.869684 

4.660957 

21 

1.860295 

2.278768 

2.785963 

3.399564 

4.140562 

5.033834 

22 

1.916103 

2.369919 

2.925261 

3.603537 

4.430402 

5.436540 

23 

1.973587 

2.464716 

3.071524 

3.819750 

4.740530 

5.871464 

24 

2.032794 

2.563304 

3.225100 

4.048935 

5.072367 

6.341181 

25 

2.093778 

2.665836 

3.386355 

4.291871 

5.427433 

6.848475 

EQUATION  OF  PAYMENTS. 


211 


340.  The  amount  of  $1  for  the  given  time  and  rate ,  multiplied 
by  the  given  principal,  gives  its  amount  for  the  same  time  and 

rate. 

Note. — When  the  interest  is  compounded  semi-annually,  it  is  com¬ 
puted  from  the  table  by  taking  one  half  the  rate  and  twice  the  number 
of  years. 

6.  What  is  the  compound  interest  of  $750  for  15  years 
at  6  %  ?  What  is  the  amount  ? 

7.  What  is  the  amount  of  $500  for  6  years  at  8%,  com¬ 
pounded  semi-annually  ? 

8.  What  is  the  amount  of  $1250  for  10  yr.  4  mo.  15  da. 
at  5%,  compound  interest? 


EQUATION  OF  PAYMENTS. 

341.  Equation  of  Payments  is  the  process  of 
finding  an  equitable  time  for  the  payment  of  several  debts, 
due  at  different  times,  without  interest.  It  is  also  called 
the  Average  of  Payments. 

The  equitable  time  sought  is  called  the  Average  Time,  or 
the  Equated  Time. 


PROBLEMS. 


1.  A  owes  B  $300,  of  which  $200  is  due  in  3  months, 
and  $100  in  6  months :  when  will  the  payment  of  $300 
equitably  discharge  the  debt  ? 


Process. 

$200X3=  $600 
$100X6=  $600 
$300  )  $1200 

4 

Ans.,  4  mos. 


A  is  entitled  to  the  use  of  $200  for  3 
months,  which  equals  the  use  of  $600  for  1 
month,  and  to  the  use  of  $100  for  6  months, 
which  equals  the  use  of  $600  for  1  month  ; 
and,  hence,  he  is  entitled  to  the  use  of  $300 
until  it  equals  the  use  of  $600  +  $600,  or 
$1200,  for  1  month.  It  will  take  $300  as  many 
months  to  equal  the  use  of  $1200  for  1  month,  as  $300  is  contained 
times  in  $1200,  which  is  4.  Hence,  the  payment  of  $300  in  4  months 
will  equitably  discharge  the  debt. 


212 


COMPLETE  ARITHMETIC. 


Proof. — In  paying  the  $200  in  4  months,  A  gains  the  use  of  $200 
for  1  month,  and  in  paying  the  $100  in  4  months,  he  loses  the  use 
of  $100  for  2  months,  which  equals  the  use  of  $200  for  1  month. 
Hence,  his  gain  and  loss  are  equal. 

2.  A  owes  a  merchant  $200  due  in  4  months,  and  $600 
due  in  8  months:  what  is  the  equated  time  for  the  pay¬ 
ment  of  both  debts? 

3.  A  owes  B  $1200,  of  which  $300  are  due  in  4  months, 
$400  in  6  months,  and  the  remainder  in  12  months :  what 
is  the  equated  time  for  the  payment  of  the  whole? 

4.  A  owes  B  $800,  of  which  is  due  in  2  months,  %  in 
3  months,  and  the  remainder  in  6  months :  what  is  the 
equated  time  for  the  payment  of  the  whole? 

5.  A  man  owes  $300  due  in  4  months,  $600  due  in  5 
months,  and  $700  due  in  10  months :  what  is  the  equated 
time  for  the  payment  of  the  whole? 

6.  Smith  &  Jones  bought  $500  worth  of  goods  on  4 
months’  credit,  $700  worth  on  6  months’  credit,  and  $1000 
worth  on  5  months’  credit:  what  is  the  equated  time  for 
the  payment  of  the  whole? 

7.  A  bought  $2000  worth  of  goods,  \  of  which  was  to  be 
paid  down,  -J-  in  3  months,  \  in  4  months,  and  the  re¬ 
mainder  in  8  months:  what  is  the  equated  time  for  the 
payment  of  the  whole? 

8.  What  is  the  equated  time  for  the  payment  of  $220, 
due  in  30  days;  $300,  due  in  40  days;  $250,  due  in  60 
days;  and  $100,  due  in  90  days? 

9.  What  is  the  equated  time  for  the  payment  of  $300, 
due  in  30  days;  $250,  due  in  45  days;  and  $350,  due  in 
60  days  ? 

Process  by  Interest. 

Int.  of  $300  for  30  days,  at  6%  =  $1.50 
Int.  of  $250  for  45  “  “6%  =1.875 
Int.  of  $350  for  60  “  “6%  =  3.50 
$900  $6,875 

$9.00  =  Int.  of  $900  for  60  days. 

.15  =  “  “  $900  for  1  day. 

$6,875  -r-  $.15  =  45.9.  Ans.,  46  days. 


EQUATION  OF  PAYMENTS. 


213 


The  debtor  is  entitled  to  the  use  (1)  of  $300  for  30  days,  which,  at 
6%,  equals  $1.50  interest;  (2)  of  $250  for  45  days,  which  equals 
$1,875  interest;  (3)  of  $350  for  60  days,  which  equals  $3.50  interest. 
Hence,  he  is  entitled  to  the  use  of  $900,  the  sum  of  the  debts,  until 
the  interest  thereon,  at  6%,  equals  the  sum  of  $1.50  +  $1,875  +  $3.50, 
which  is  $6  875.  The  interest  of  $900  for  1  day  is  $.15;  and  since 
$6,875  -j-  $.15  =  45.9,  it  will  take  45.9  days  for  $900  to  yield  $6,875 
interest.  The  equated  time  for  payment  is  46  days. 

Note. — When  the  fraction  of  a  day  in  the  equated  time  is  more 
than  it  is  counted  as  a  day ;  when  it  is  less  than  £,  it  is  disre¬ 
garded. 

10.  What  is  the  equated  time  for  the  payment  of  $520, 
due  in  45  days ;  $340,  due  in  60  days ;  and  $640,  due  in 
90  days  ? 

11.  What  is  the  equated  time  for  the  payment  of  $375, 
due  now ;  $425,  due  in  30  days ;  $500,  due  in  60  days ; 
and  $600,  due  in  75  days? 

12.  What  is  the  equated  time  for  the  payment  of  $340, 
due  May  10,  1870;  $450,  due  June  10;  $560,  due  July 
15 ;  and  $650,  due  Aug.  10  ? 

Note. — Begin  with  the  first  date  (May  10),  and  find  the  exact 
number  of  days  between  it  and  each  succeeding  date.  The  equated 
time  is  counted  forward  from  the  first  date. 

13.  What  is  the  equated  time  for  the  payment  of  $1000, 
due  June  1,  1870;  $850,  due  July  1;  $750,  due  Sept.  1; 
and  $900,  due  Oct.  1  ? 

14.  What  is  the  equated  time  for  the  payment  of  $75, 
due  May  6,  1870;  $115,  due  May  26;  $220,  due  June  25; 
$315,  due  July  16;  and  $350,  due  July  30? 

PRINCIPLES  AND  RULES. 

342.  The  time  between  the  contraction  of  a  debt  and  its 
payment  is  called  the  Term  of  Credit,  or  Time  of  Credit . 

343.  Principles. — 1.  The  payment  of  a  sum  of  money 
before  it  is  due  is  offset  by  keeping  an  equal  sum  of  money 
an  equal  time  after  it  is  due. 

2.  The  use  of  any  sum  of  money  is  measured  by  its  interest 
for  the  time . 


214 


COMPLETE  ARITHMETIC. 


344.  Rules. — To  equate  the  time  of  several  debts  or 
payments, 

1.  Multiply  each  debt  or  payment  by  its  time  of  credit,  and 
divide  the  sum  of  the  products  by  the  sum  of  the  debts  or  pay¬ 
ments.  Or, 

2.  Compute  the  interest  of  each  debt  or  payment  for  its  time 
of  credit,  and  divide  the  sum  of  the  interests  by  the  interest  of 
the  sum  of  the  debts  or  payments  for  one  month  or  one  day. 

Notes. — 1.  As  the  result  will  be  the  same  at  any  rate,  the  interest 
may  be  computed  at  that  rate  which  is  most  convenient. 

2.  The  correctness  of  each  of  the  above  methods  has  been  called 
in  question  by  a  number  of  authors,  who  commend  the  following  as 
“  the  only  accurate  rule  ” : 

“ Find  the  present  worth  of  each  of  the  given  amounts  due  ;  then  find  in 
what  time  the  sum  of  these  present  worths  will  amount  to  the  sum  of  all  the 
payments. 

The  inaccuracy  of  this  so-called  “accurate  rule”  is  easily  shown. 
The  methods  given  above  are  both  strictly  accurate,  and  they  are  in 
general  use.  (See  appendix.) 

345.  When  partial  payments  are  made  on  a  debt  before 
it  is  due,  the  time  for  the  payment  of  the  balance  of  the 
debt  is  proportionately  extended. 

15.  A  owes  a  merchant  $200,  due  in  12  months,  without 
interest;  in  4  months  he  pays  $50  on  the  debt,  and  in  8 
months,  $50 :  when  in  equity  should  he  pay  the  balance  ? 


Process. 

$50  X  8  =  $400 
$50  X  4  =  $200 


$200  —  $100  =  $100  )  $600 

6 


In  paying  $50  in  4  months,  A  loses 
its  use  for  8  months,  and  in  paying 
$50  in  8  months,  he  loses  its  use  for  4 
months,  and  hence  he  loses  the  use  of 
$400  -(-  $200,  or  $600,  for  1  month.  To 
offset  this  loss,  he  is  entitled  to  keep 


the  balance  ($100)  6  months  after  its  maturity. 


16.  A  owes  B  $300,  due  in  8  months :  if  he  pay  $200 
in  5  months,  when  should  he  pay  the  balance? 

17.  A  man  bought  a  horse,  agreeing  to  pay  $150  in  6 


EQUATION  OF  ACCOUNTS. 


215 


months,  without  interest:  if  he  pay  $50  down,  when  should 
he  pay  the  balance? 

18.  A  owes  B  $600,  payable  in  6  months,  but,  at  the  close 
of  3  months,  he  proposes  to  make  a  payment  sufficiently 
large  to  extend  the  time  for  the  payment  of  the  balance  6 
months.  How  large  a  payment  must  he  make  ? 

19.  A  owed  B  $1500,  due  in  12  months,  but  in  4  months 
paid  him  $400,  and  in  6  months  $500 :  when  in  equity  ought 
the  balance  to  be  paid? 

20.  Clark  and  Brown  bought  March  10,  1870,  a  bill  of 
goods  amounting  to  $2500,  on  4  months’  credit ;  but  they 
paid  $650  Apr.  7 ;  $500  Apr.  30 ;  and  $350  May  20. 
When  ought  they  to  pay  the  balance? 

346.  Rule. — Multiply  each  payment  by  the  time  it  was  paid 
before  it  was  due ,  and  divide  the  sum  of  the  products  by  the  bal¬ 
ance  unpaid. 


EQUATION  OF  ACCOUNTS. 

Note. — Equation  of  Accounts,  especially  Case  II,  needs  to  be 
studied  only  by  advanced  classes.  It  is  seldom  used  in  ordinary 
business. 

347.  Equation  of  Accounts  is  the  process  of 
finding  the  equated  time  for  the  payment  of  the  balance  of 
an  account,  or  the  time  when  the  balance  was  due. 

Case  I. 

Accounts  containing  only  Debit  Items. 

1.  A  bookseller  bought  of  Wilson,  Hinkle  &  Co.  the 
following  bills  of  goods,  on  4  months’  credit: 

Feb.  3,  1870,  a  bill  of  $450. 

“  24,  “  “  500. 

Mch.  25,  “  “  750. 

Apr.  20,  “  “  600. 

What  is  the  equated  time  of  maturity? 


216 


COMPLETE  ARITHMETIC. 


Process. 

Dae  June  3,  1870,  $450  X  00  = 

“  “  24,  “  500  X  21  =  10500 

“  July  25,  “  750  X52  =  39000 

“  Aug.  20,  “  600  X  78  =  46800 

$2300  )  $96300  (  41.8  days. 

The  equated  date  of  maturity  of  the  above  bills  is  42  days  from 
June  3,  1870,  which  is  July  15,  1870. 

Notes. — 1.  The  date  of  maturity  of  each  bill  is  found  by  counting 
forward  4  months  from  the  date  of  purchase.  The  same  result  would 
be  obtained  by  finding  the  average  or  equated  date  of  purchase,  and 
counting  forward  4  months. 

2.  The  equated  time  of  maturity  may  also  be  found  by  beginning 
at  the  last  date,  and  taking  the  exact  number  of  days  between  each 
preceding  date  and  the  last  date  for  a  multiplier.  The  equated  date 
is  then  found  by  counting  back  from  the  last  date. 

2.  Murray  &  Co.  bought  of  Smith  &  Moore  goods  as 
follows : 

Apr.  15,  1869,  a  bill  of  $400,  on  3  mo.  credit. 

May  20,  “  “  245,  on  4  “  “ 

June  25,  “  “  375,  on  4  “  “ 

Sept.  15,  “  “  625,  on  3  “  “ 

What  is  the  equated  time  of  maturity  ? 

3.  A  merchant  has  the  following  charges  against  a  cus¬ 
tomer  : 

May  9,  1870,  $340,  on  4  mo.  credit. 

June  6,  “  530,  on  4  “  “ 

July  8,  “  213,  on  3  “  “ 

Aug.  30,  “  150,  on  4  “  “ 

What  is  the  equated  time  of  maturity? 

4.  J.  O.  Bates  &  Co.  bought  of  Smith  &  Brown  several 
bills  of  goods,  as  follows : 


March 

3, 

1868, 

a  bill  of  $250,  on  3 

mo.  credit. 

April 

15, 

<< 

u 

180,  on  4 

a  a 

June 

20, 

(< 

it 

325,  on  3 

it  a 

Aug. 

10, 

<( 

a 

80,  on  3 

((  u 

Sept. 

1, 

<< 

a 

100,  on  4 

a  a 

What  is  the  equated  date  of  maturity?  How  much  would 
pay  the  account  Dec.  1,  1868? 


EQUATION  OF  ACCOUNTS. 


217 


348.  Rule. — To  find  the  equated  time  maturity  for 
the  debit  items  of  an  account,  First  find  th  /  maturity  of  each 
item  or  bill,  and  then,  counting  from  the  first  date  for  the  time 
of  credit ,  find  the  equated  time  as  in  the  equation  of  payments. 
The  date  of  the  equated  time  is  found  by  counting  forward  from 
the  first  date. 

Notes. — 1.  The  equated  time  may  be  found  by  interest,  as  in  the 
Equation  of  Payments.  (Art.  344,  Rule  2.) 

2.  The  sum  of  the  debit  items  draws  interest  from  the  equated  date 
of  maturity  to  the  date  of  payment. 

Case  II. 

Accounts  containing  "both  Debits  and.  Credits. 

5.  What  is  the  equated  date  of  maturity  of  each  side  of 
the  foliowing  account? 

Dr.  John  Smith  in  account  with  John  Jones.  Or. 


1868. 

Time  of  Cred. 

1868. 

Apr.  3, 

To  Mdse. 

$220 

3  mo. 

July  1, 

By  Cash 

$200 

June  1, 

U 

125 

4  “ 

Oct.  3, 

« 

150 

July  15, 

a 

200 

4  “ 

Dec.  20, 

u 

300 

Aug.  24, 

u 

140 

6  “ 

Oct.  1, 

u 

190 

6  “ 

Process. 


Debits. 

Due 

July  3, 1 868,  $220  X  00  = 

Oct.  1,  “  125  X  90  =  11250 

Nov.  15,  “  200  X  135  =  27000 

Feb.  24,1869,  140X236  =  33040 
Apr.  1,  “  190  X  272  =  51680 

$875  )  $122970 

141 

Debits  are  due  141  days  from 
July  3,  1868,  which  is  Nov.  21. 


Credits. 

Due 

July  1, 1868,  $200  X  00  = 

Oct.  3,  “  150  X  94  =  14100 

Dec.  20,  “  300  X  172  =  51600 

$650  )  $65700 

101 

Credits  are  due  101  days  from 
July  1,  which  is  Oct.  10. 


Note. — Each  side  of  the  account  may  be  equated  without  refer¬ 
ence  to  the  other,  as  is  done  above,  or  the  first  or  last  date  of  the  ac¬ 
count  may  be  made  a  common  starting-point  for  both  sides. 

C.Ar.— 19. 


218 


COMPLETE  ARITHMETIC. 


6.  The  above  account,  as  equated,  stands  thus : 

Dr.  Cr. 

Due  Nov.  21,  1868  .  .  $875  Due  Oct.  10,  1868  .  .  $650 

When  is  the  balance  of  the  account  due? 


Process. 

Debits . $875 

Credits . 650 

Balance . $225 

Difference  in  time,  42  days. 

Balance  is  due  121  days  from  Nov.  21,  1868,  which  is  March  22, 
1869. 

Suppose  the  account  settled  Nov.  21,  the  later  date.  Since  the 
credit  side  of  the  account  has  been  due  since  Oct.  10,  it  has  been  draw¬ 
ing  interest  for  42  days.  To  increase  the  debit  side  of  the  account 
by  an  equal  amount  of  interest,  the  balance  must  remain  unpaid  121 
days.  Counting  forward  121  days  from  Nov.  21,  the  balance  is  found 
to  be  due  March  22,  1869. 


$650 
_ 42 

$225  )  $27300 
121 


7.  Suppose  that  the  debit  and  credit  sides  of  an  account 
when  equated  stand  as  follows: 

Dr.  Or. 

Due  Nov.  21,  1868  .  .  $650  Due  Oct.  10,  1868  .  .  $875 

What  would  be  the  equated  time  of  payment  for  the  bal¬ 
ance  ? 

Process. 


Credits . $875 

Debits . 650 

Balance . $225 


Difference  in  time,  42  days. 


$875 
_ 42 

$225 )  $36750 
163 


Balance  is  due  163  days  prior  to  Nov.  21,  1868,  which  is  June  11, 
1868. 

Suppose  the  account  settled  Nov.  21,  as  before.  The  credit  side, 
having  been  due  since  Oct.  10,  has  been  drawing  interest  for  42  days. 
That  the  debit  side  of  the  account  may  be  increased  by  an  equal 
amount  of  interest,  the  balance  must  be  regarded  as  due  163  days 
prior  to  Nov.  21. 


EQUATION  OF  ACCOUNTS. 


219 


8.  The  debit  and  credit  sides  of  an  account  when  equated 
stand  as  follows : 

Dr.  Cr. 

Due  June  5,  1870  .  .  $1285  Due  July  1,  1870  .  .  $1000 

What  is  the  equated  time  of  payment  for  the  balance  ? 


9.  At  what  time  did  the  balance  of  the  following  equated 
account  begin  to  draw  interest : 

Dr.  Cr. 

Due  July  12,  1870  .  .  $450  Due  Sept.  1,  1870  .  .  $800 


10.  When  will  the  balance  of  the  following  account 
begin  to  draw  interest,  the  debit  items  having  a  credit  of  3 
months  ? 


Dr.  It.  Hill  &  Co.,  in  account  with  O.  Cooke.  Cr. 


1870. 

1870. 

July  10 

To  Mdse. 

$120 

Nov.  20 

By  Cash 

$350 

“  30 

a 

450 

Dec.  25 

“  Mdse. 

250 

Aug.  30 

(( 

380 

1871. 

Sept.  9 

(( 

560 

“  30 

u 

400 

Jan.  1 

“  Cash 

750 

349.  Rule. — To  find  the  equated  time  for  the  payment 
of  the  balance  of  an  account, 

1.  Find  the  equated  time  for  each  side  of  the  account. 

2.  Multiply  the  side  of  the  account  which  falls  due  first  by 
the  number  of  days  between  the  dates  of  the  equated  time  of  the 
two  sides,  and  divide  the  product  by  the  balance  of  the  account. 

3.  The  quotient  will  be  the  number  of  days  to  the  maturity  of 
the  balance,  to  be  counted  forward  from  the  later  equated  date 
when  the  smaller  side  of  the  account  falls  due  first,  and  back¬ 
ward  when  the  larger  side  falls  due  first. 

Notes. — 1.  When  an  account  is  settled  by  cash,  each  side  of  the 
account  is  increased  by  its  interest  from  maturity  to  the  date  of  set¬ 
tlement,  and  the  difference  between  the  two  sides  thus  increased  by 
interest,  is  called  the  Cash  Balance.  Instead  of  adding  the  accrued 
interest  to  each  side,  the  balance  of  interest  may  be  found  and  added 


220 


COMPLETE  ARITHMETIC. 


to  or  subtracted  from  the  balance  of  items,  according  as  the  two  bal¬ 
ances  fall  upon  the  same  or  upon  opposite  sides  of  the  account.  Thus, 
in  problem  6  above,  the  balance  of  interest,  which  is  the  interest  of 
$650  for  42  days,  falls  on  the  credit  side,  and  the  balance  of  items  on 
the  debit  side.  The  cash  balance  is  $225  —  $3.90,  which  is  $221.10. 

2.  The  cash  balance  may  be  found  directly,  without  equating  the 
account,  by  finding  the  interest  of  each  item  from  its  maturity  to 
the  date  of  settlement,  and  taking  the  difference  between  the  sums 
of  the  debit  interests  and  credit  interests  for  the  balance  of  interest. 
When  the  balance  of  interest  and  the  balance  of  items  fall  on  the 
same  side,  the  cash  balance  is  their  sum;  when  they  fall  on  opposite 
sides,  the  cash  balance  is  their  difference. 


SECTION  XV. 

RATIO  AND  PROPORTION. 

RATIO. 

350.  The  relation  between  two  numbers  expressed  by 
their  quotient,  is  called  their  Ratio.  The  ratio  of  6  to  2  is 
6  a-  2,  or  3 ;  and  the  ratio  of  2  to  6  is  2  —  6,  or 

MENTAL  EXERCISES. 

1.  What  is  the  ratio  of  8  to  4?  24  to  8?  45  to  15? 

2.  What  is  the  ratio  of  6  to  12  ?  12  to  36?  16  to  64? 

3.  What  is  the  ratio  of  42  to  14?  14  to  42?  12  to  30? 

4.  What  is  the  ratio  of  50  to  15?  15  to  50?  80  to  25? 

5.  What  is  the  ratio  of  36  to  16  ?  60  to  25  ?  70  to  40  ? 

6.  What  is  the  ratio  of  45  to  60  ?  18  to  45  ?  75  to  45  ? 

7.  What  is  the  ratio  of  $33  to  $11?  $20  to  $50?  $45 
to  $36?  $50  to  $150? 

8.  What. is  the  ratio  of  16  lb.  to  40  lb.?  28  lb.  to  13  lb.? 

9.  What  is  the  ratio  of  T9^  to  Ts7?  T3y  to  -j^-?  yf  to  -^-? 

10.  What  is  the  ratio  of  \  to  J?  y  to  J?  £  to  ^? 

11.  What  is  the  ratio  of  J  to  |?  §•  to  §?  f  to  f  ? 

12.  What  is  the  ratio  of  5  to  J-?  \  to  4?  }  to  2^-? 


RATIO. 


221 


WHITTEN  EXERCISES. 


351.  The  ratio  of  two  numbers  is  expressed  by  placing  a 
colon  between  them.  The  ratio  of  4  to  10  is  denoted  by 

4  :  10,  and  the  ratio  of  |  to  |  by  |  The  expression 

4  :  10  is  read  the  ratio  of  4  to  10. 

13.  Express  the  ratio  of  7  to  15.  12  to  35.  35  to  17. 

14.  Express  the  ratio  of  2.5  to  7.5.  3.4  to  .62. 

15.  Express  the  ratio  of  f  to  -§.  f  to  5^.  2^  to  f. 

16.  What  is  the  value  of  the  ratio  of  112  to  35? 


Process  :  112  :  35  =  112  -s-  35  =  3b  Ans. 


What  is  the  value  of 

17.216:81?  21.  -A-':  |?  25.  6  qt. :  3  pk.  ? 

18.  129  :  215  ?  22.  150 : 16} ?  26.  5  lb.  12  oz. :  17  lb.  4  oz.? 

19.  14.3:6.5?  23.  12^:30^?  27.  2  ft.  6  in. :  12  ft.  6  in.  ? 

20.  1.44 : 3.2  ?  24.  34^ :  5J  ?  28.  15  pk. :  12  bu.  2  pk.  ? 

29.  Reduce  24  :  60  to  its  lowest  terms. 

Process  :  24  :  60  =  ■§$  =  $  =  2  :  5,  Ans. 

Reduce  the  following  ratios  to  their  lowest  terms: 


30.  35  :  84. 

31.  63  :  108. 

32.  121  :  220. 


33.  105  :  140. 

34.  81  :  189. 

35.  105  :  195. 


36.  169  :  65. 

37.  256  :  112. 

38.  225  :  120. 


39.  Reduce  £  :  f-  to  an  equal  ratio  with  integral  terms. 
Process  :  f  :  £  =  :  t9j  =  10  :  9. 

Reduce  the  following  ratios  to  equal  ratios  with  integral 
terms  : 


40. 

2 

J 

.  3 
•  ¥* 

43. 

7  •  5 
¥  *  ¥• 

46. 

i-.io. 

41. 

5 

T 

.  1  1 
.  x?. 

44. 

J1  •  7 

TW  *  TJ' 

47. 

Ol  .  5 
■“3  *  6* 

42. 

TT 

.  5 

•  8* 

45. 

13  .  17 

1  5  •  Tor* 

48. 

14  :  5}. 

49.  Multiply  10  :  21  by  14  :  15. 


Process-  I  10:21  —  $}.  14:15  —  ff. 

*  l  Hence,  (10  :  21)  X  (14  :  15)  =  X  =  f,  Ans. 


222 


COMPLETE  ARITHMETIC. 


50.  What  is  the  product  of  9  :  10  and  24:33? 

51.  What  is  the  product  of  7  :  15,  25  :  14,  and  24  :  35? 

52.  What  is  the  product  of  12  :  25,  15  :  24,  and  16  :  21  ? 

DEFINITIONS,  PRINCIPLES,  AND  RULES. 

352.  j Ratio  is  the  relation  between  two  numbers  of  the 
same  kind  expressed  by  their  quotient. 

353.  The  two  numbers  compared  are  called  the  Terms 
of  the  ratio. 

The  first  term  is  the  Antecedent,  and  the  second  term  the 
Consequent.  The  two  terms  form  a  Couplet. 

354.  The  value  of  a  ratio  is  the  quotient  obtained  by 
dividing  the  antecedent  by  the  consequent. 

When  the  antecedent  is  greater  than  the  consequent,  the  value  of 
the  ratio  is  greater  than  1 ;  when  the  antecedent  is  less  than  the  con¬ 
sequent,  the  value  is  less  than  1. 

355.  The  ratio  of  two  numbers  is  expressed  by  placing  a 
colon  (:)  between  them;  as,  5:12.  The  colon  is  called 
the  Sign  of  Ratio. 

Note. — The  sign  of  ratio  is  the  sign  of  division  with  the  hori¬ 
zontal  line  omitted. 

356.  A  ratio  is  also  expressed  in  the  form  of  a  fraction, 
the  antecedent  being  made  the  numerator  and  the  conse¬ 
quent  the  denominator.  Thus,  5  :  12  =  T\. 

Note. — Several  American  authors  divide  the  consequent  by  the 
antecedent,  thus  reversing  the  positions  of  dividend  and  divisor,  as 
indicated  by  the  sign  of  division.  The  great  majority  of  mathe¬ 
matical  writers  divide  the  antecedent  by  the  consequent. 

Ratios  are  either  Simple  or  Compound. 

357.  A  Simple  Hatio  is  the  ratio  of  two  numbers; 

as  5  :  8,  or  f  :  f . 

Note. — A  simple  ratio,  having  one  or  both  of  its  terms  fractional, 
is  called  by  several  authors  a  Complex  Ratio. 


RATIO. 


223 


358.  A  Compound  Ratio  is  the  product  of  two  or 
more  simple  ratios ;  as,  (5  :  6)  X  ($  :  10). 

It  may  be  expressed  in  three  ways,  as  follows : 

(5  :  6)  X  (8  :  9)  X  (f  :  10) ;  or^X^Xfj  or  8  :  9. 

6  9  10  |  .  10. 

359.  An  Invevse  Ratio  is  a  ratio  resulting  from  an 
inversion  of  the  terms  of  a  given  ratio.  Thus,  5  :  7  is  the 
inverse  of  7  :  5.  It  is  also  called  a  Reciprocal  Ratio. 

360.  Principles. — 1.  The  two  terms  of  a  ratio  must  be  like 
numbers. 

2.  The  antecedent  equals  the  consequent  multiplied  by  the 
ratio. 

3.  The  consequent  equals  the  antecedent  divided  by  the  ratio. 

4.  If  the  product  of  the  two  terms  of  a  ratio  be  divided  by 
either  term ,  the  quotient  will  be  the  other  term. 

5.  A  ratio  is  multiplied  by  multiplying  the  antecedent  or 
dividing  the  consequent  by  a  number  greater  than  1. 

6.  A  ratio  is  divided  by  dividing  the  antecedent  or  multiplying 
the  consequent  by  a  number  greater  than  1. 

7.  A  ratio  is  not  changed  by  multiplying  or  dividing  both  of 
its  terms  by  the  same  number. 

8.  The  product  of  two  or  more  ratios  equals  the  ratio  of  their 
products. 

361.  Rules. — 1.  To  reduce  a  simple  ratio  to  its  lowest 
terms,  Divide  both  terms  by  their  greatest  common  divisor. 
(Pr.  7.) 

2.  To  reduce  a  simple  ratio  with  fractional  terms  to  one 
with  integral  terms,  Multiply  both  terms  by  the  least  common 
multiple  of  the  denominators  of  the  fractions.  (Pr.  7.) 

o.  To  find  the  product  of  two  or  more  simple  ratios, 
Midtiply  the  antecedents  together  for  an  antecedent  and  the  con¬ 
sequents  together  for  a  consequent.  (Pr.  8.) 

Note. — The  process  may  be  shortened  by  cancellation. 


224 


COMPLETE  ARITHMETIC. 


PROPORTION. 

MENTAL  EXERCISES. 

The  ratio  of  12  to  6  is  equal  to  the  ratio  of  14  to  7, 
since  the  value  of  each  ratio  is  2. 

1.  What  two  numbers  have  a  ratio  to  each  other  equal 
to  the  ratio  of  15  to  5  ?  24  to  12? 

2.  What  two  numbers  have  a  ratio  to  each  other  equal 
to  6:  24?  7:21?  11:44? 

3.  What  two  numbers  have  a  ratio  to  each  other  equal 
to  45:  15?  12:60?  72:24? 

4.  To  what  number  has  10  a  ratio  equal  to  the  ratio 
of  30  to  15?  14  to  28? 

5.  To  what  number  has  16  a  ratio  equal  to  11  :33? 

6.  To  what  number  has  12  a  ratio  equal  to  6:30? 
24:16?  20  to  15? 

7.  12  is  to  60  as  5  is  to  what  number? 

8.  13  is  to  39  as  15  is  to  what  number? 

9.  14  is  to  42  as  25  is  to  what  number  ? 

10.  56  is  to  8  as  63  is  to  what  number? 

362.  The  equality  of  two  ratios  is  expressed  by  placing  a 

double  colon  ( : : )  between  them.  Thus,  5  :  10  =  7  :  14  is 
written  5  :  10  : :  7  :  14,  and  is  read  5  is  to  10  as  7  is  to  14. 

11.  Read  8  :  40  : :  12  :  60,  and  show  that  the  two  ratios 

are  equal. 

12.  Read  27  :  9  : :  63  :  21,  and  show  that  the  two  ratios 
are  equal. 

13.  Read  5  :  2J  : :  25  :  12J,  and  show  that  the  two  ratios 
are  equal. 

DEFINITIONS  AND  PRINCIPLES. 

363.  A  _P ropovtion  is  an  equality  of  ratios. 

364.  The  first  ratio  of  a  proportion  is  called  the  First 
Couplet,  and  the  second  ratio  the  Second  Couplet. 


SIMPLE  PROPORTION. 


225 


365.  The  first  and  third  terms  of  a  proportion  are  the 
Antecedents,  and  the  second  and  fourth  terms  the  Conse¬ 
quents. 

Note. — The  antecedents  of  a  proportion  are  the  antecedents  of  its 
ratios,  and  the  consequents  are  the  consequents  of  its  ratios. 

366.  The  first  and  fourth  terms  of  a  proportion  are  the 
Extremes,  and  the  second  and  third  terms,  the  Means. 

The  four  terms  of  a  proportion  are  called  Proportionals,  and  the 
last  is  the  fourth  proportional  to  the  other  three  in  their  order. 

367.  Three  numbers  are  in  proportion  when  the  ratio  of 
the  first  to  the  second  equals  the  ratio  of  the  second  to  the 
third ;  as,  8  :  12  : :  12  :  18.  The  second  number  is  called  a 
mean  'proportional. 

368.  Proportions  are  either  Simple  or  Compound. 

A  Simple  Proportion  is  an  equality  of  two  simple 
ratios. 

A  Compound  Proportion  is  an  equality  of  two 
ratios,  one  or  both  of  which  are  compound. 

* 

SIMPLE  PROPORTION. 

Case  I. 

Any  Term  found,  when  the  other  Three  Terms 

are  given. 

369.  The  proportion  4  :  8  : :  6  :  12  may  be  written  4  :  8 
=  6  :  12,  or  f  =  1^-  (Art.  356)  ;  and  multiplying  the  two 
equal  fractions  by  12  and  8,  their  denominators,  we  have 
4X12  =  6x8.  Hence,  the  following 

Principles. — 1.  The  product  of  the  extremes  of  a  propor¬ 
tion  equals  the  product  of  the  means.  Hence, 

2.  If  the  product  of  the  extremes  of  a  proportion  be  divided 
by  either  mean,  the  quotient  will  be  the  other  mean. 

3.  If  the  product  of  the  two  means  of  a  proportion  be  divided 
by  either  extreme,  the  quotient  will  be  the  other  extreme. 


226 


COMPLETE  ARITHMETIC. 


WRITTEN  PROBLEMS. 


Find  the  missing  term  in  the  following  proportions : 


14.  21  :  7  : : 

15.  15  :  40 

16.  —  :  24 

17.  —  :  9  : 

18.  45  :  30 

19.  2.5:62.5 

20.  7.2  :  —  : 

21.  .25  :  —  : 


36:  — 

:  18  :  — 

:  8  :  32 
60:  18 
:  —  :  24 
:  — :  3.25 
4.7  :  9.4 
2.5  :  7.5 


99  _2.  .  3  »  .  5  .  

•  Q  •  A  •  •  ft  • 


3 

OQ  3.2..  _  .  Z 

40.  j  .  -g-  .  .  .  f 

24.  —  :  24 


¥ 

2 


1 

¥ 


3 

T 


95  i.  .  . .  x  .  l 

•  3  *  #  *  5  # 

26.  $5  :  $45  : :  6  lb. 

27.  $.75  :  $3  : :  —  : 


56  oz. 


28.  16  men  :  96  men  ::  15  days  :  — 

29.  8  horses  :  14  horses  ::  f  :  — 


370.  Rules. — 1.  To  find  either  extreme  of  a  simple  pro¬ 
portion,  Divide  the  product  of  the  two  means  by  the  other 
extreme. 

2.  To  find  either  mean  of  a  simple  proportion,  Divide  the 
product  of  the  two  extremes  by  the  other  mean. 


Case  II. 

The  Solution  of  Problems  by  Simple  Proportion. 

371.  The  solution  of  a  problem  by  proportion  consists  of 
two  parts,  viz. : 

1.  The  arranging  of  the  three  given  terms,  called  the 
Statement. 

2.  The  finding  of  the  fourth  term  by  Case  I. 

372.  If  the  required  answer  be  made  the  fourth  term  of 
a  proportion,  the  given  number  of  the  problem,  which  is  of 
the  same  kind  as  the  answer,  will  be  the  third  term,  since 
the  two  terms  of  a  ratio  must  be  like  numbers.  (Art.  360.) 

373.  Of  the  two  remaining  numbers  given  in  the  problem, 
the  greater  will  be  the  second  term  when  the  answer  is  to 
be  greater  than  the  third  term,  and  the  less  will  be  the 
second  term  when  the  answer  is  to  be  less  than  the  third 
term,  otherwise  the  two  ratios  can  not  be  equal. 


SIMPLE  PROPORTION. 


227 


WHITTEN  PROBLEMS. 


30.  If  15  yards  of  cloth  cost  $24,  what  will  40  yards 
cost? 

Since  the  cost  of  40  yards  is  to  be 
the  answer,  make  $24,  the  cost  of  15 
yards,  the  third  term  of  a  proportion ; 
and  since  40  yards  will  cost  more  than 
15  yards,  the  fourth  term  is  to  be  greater 
than  the  third,  and  hence  the  second 
term  must  be  greater  than  the  first.  Make  40  yards  the  second  term 
and  15  yards  the  first,  giving  the  proportion  15  yd.  :  40  yd. : :  $24  :  cost 
of  40  yards,  which,  by  Case  I,  is  found  to  be  $64. 


STATEMENT. 

15  yd.  :  40  yd.  ::  $24  : 

40 

15)  $960 
$64, 


Arts. 


Ans. 


31.  If  45  sheep  cost  $565,  what  will  140  sheep  cost? 

32.  If  13  tons  of  hay  cost  $97.50,  what  will  7J  tons 
cost? 

33.  If  70  acres  of  land  cost  $1875,  what  will  320  acres 
cost? 

34.  If  120  acres  of  land  cost  $3000,  how  many  acres  can 
be  bought  for  $4500? 

35.  If  4  lb.  6  oz.  of  butter  cost  $1.75,  what  will  17J 
pounds  cost? 

36.  If  a  man’s  pulse  beat  75  times  in  a  minute,  how 
many  times  will  it  beat  in  8  hours  ? 

37.  If  a  clock  ticks  120  times  in  a  minute,  how  many 
times  does  it  tick  in  9 J-  hours  ? 

38.  If  a  comet  move  4°  20'  in  15  hours,  how  far  will  it 
move  in  5  days? 

39.  If  a  garrison  of  160  men  consume  24  barrels  of  flour 
in  6  weeks,  how  many  barrels  will  supply  it  one  year? 

40.  If  24  barrels  of  flour  will  supply  160  men  6  weeks, 
how  many  barrels  will  supply  360  men  the  same  time? 

41.  If  a  vertical  staff  3  feet  high  casts  a  shadow  5  feet 
long,  how  long  a  shadow  will  a  pole  120  feet  high  cast  at 
the  same  time? 

42.  If  a  pole  20  feet  high  casts  a  shadow  12  feet  long, 
how  high  is  the  tree  whose  shadow,  at  the  same  time,  is  90 
feet  long? 


228 


COMPLETE  ARITHMETIC. 


43.  If  f  of  a  farm  is  worth  $4500,  what  is  f  of  it  worth? 

44.  If  of  a  yard  of  silk  cost  $2.10,  what  will  16|  yards 
cost? 

45.  If  6J  tons  of  hay  cost  $58 ,75,  how  many  tons  can  be 
bought  for  $173.90? 

46.  At  the  rate  of  5  peaches  for  8  apples,  how  many 
apples  can  be  bought  for  5  dozen  peaches? 

47.  If  12  men  can  mow  20  acres  of  grass  in  a  day,  how 
many  acres  can  25  men  mow  ? 

48.  If  9  men  can  build  a  wall  in  15  days,  how  long  will 
it  take  5  men  to  build  it? 


The  15  days  is  the  third  term, 
since  the  answer  is  to  be  in  days. 
If  it  take  9  men  15  days  to  build 
a  wall,  it  will  take  5  men  more 
than  15  days,  and  hence  the  an¬ 
swer,  or  fourth  term,  is  greater 
than  the  third  term,  and  consequently  the  second  term  must  be  greater 
than  the  first  term.  The  proportion  is  5  men  :  9  men  : :  15  days  :  27 
days. 


STATEMENT. 

5  men  :  9  men  : :  15  days  :  Ans. 
_ 9 

5)135 

27  days,  Ans. 


Note. — The  principle  involved  in  this  class  of  problems  may  thus 
be  stated :  The  greater  the  cause,  the  less  the  time  required  to  “produce  a 
given  effect ;  and,  conversely,  the  greater  the  time,  the  less  the  cause  required. 


49.  If  a  quantity  of  provisions  will  supply  a  garrison  of 
90  men  125  days,  how  long  will  the  same  provisions  supply 
150  men? 

50.  If  15  men  can  harvest  a  field  of  wheat  in  12  days, 
how  many  men  can  harvest  it  in  5  days? 

51.  Divide  90  into  two  parts  whose  ratio  is  equal  to  the 
ratio  of  4  and  5. 


Proportions, 


{ 


(4  -{-  5)  :  90  : :  4  :  Smaller  part. 
(4  +  5)  ;  90  : :  5  :  Greater  part. 


Note. — These  proportions  are  based  on  the  principle  that  when 
four  numbers  are  in  proportion,  the  sum  of  the  first  and  second  terms  is 
to  the  sum  of  the  third  and  fourth  terms  as  the  first  term  is  to  the  third,  or 
as  the  second  term  is  to  the  fourth. 


52.  Divide  640  into  two  parts  proportional  to  8  and  12. 
To  9  and  11. 


SIMPLE  PROPORTION. 


229 


53.  An  estate  worth  $9600  was  divided  between  two  heirs 
in  proportion  to  their  ages,  which  were  15  years  and  17  years 
respectively:  how  much  did  each  receive? 

54.  Two  men,  150  miles  apart,  are  approaching  each 
other,  one  traveling  2  miles  to  the  other  3 :  how  far  will 
each  travel  before  they  meet? 

6^“For  additional  problems  see  Problems  for  Analysis  p.  239. 


PRINCIPLES  AND  RULE. 


374.  Principles. — 1.  The  ratio  of  two  like  causes  equals 
the  ratio  of  their  effects.  Conversely, 

2.  The  ratio  of  two  like  effects  equals  the  ratio  of  their  causes. 

3.  The  ratio  of  two  like  causes  equals  the  inverse  ratio  of 
their  tunes.  Conversely, 

4.  The  ratio  of  the  times  of  two  like  causes  equals  the  in¬ 
verse  ratio  of  the  causes. 

5.  The  two  terms  of  each  couplet  of  a  simple  proportion  must 
he  like  numbers.  (Art.  360,  Pr.  1.) 

6.  The  fourth  term  of  a  'proportion  equals  the  product  of  the 
second  and  third  terms  divided  by  the  first  term.  (Art.  370.) 

375.  Rule. — To  solve  a  problem  by  simple  proportion, 

1.  Take  for  the  third  term  the  number  which  is  of  the  same 
kind  as  the  answer  sought ,  and  make  the  other  two  numbers  the 
first  couplet ,  placing  the  greater  for  the  second  term ,  when  the 
answer  is  to  be  greater  than  the  third  term;  and  the  less  for 
the  second  term ,  when  the  answer  is  to  be  less  than  the  third  term. 

2.  Divide  the  product  of  the  second  and  third  terms  by  the 
first  term,  and  the  quotient  will  be  the  fourth  term,  or  answer. 

Notes. — 1.  When  the  terms  of  the  first  couplet  are  denominate 
numbers,  they  must  be  reduced  to  the  same  denomination. 

2.  The  process  of  finding  the  fourth  term  may  be  shortened  by  can¬ 
cellation.  The  proportion  15  :  45  : :  27.5  :  —  may  be  completed  thus: 


27.5 

U 


=  82.5. 


4$3 

27.5 


27.5X3  =  82.5 


3.  The  process  of  solving  problems  by  simple  proportion  is  also 
called  “  The  Ride  of  Three.” 


230 


COMPLETE  ARITHMETIC. 


COMPOUND  PROPORTION. 


•  Case  I. 

RecLiaction  of  Compound  Ratios  and  Proportions 

to  Simple  Ones. 


1.  Reduce  the  compound  ratio 


ratio  in  its  lowest  terms. 

Process. 


to  a  simple 


Or: 


20 

4 

6 


80 

3 

8 


20X4X6:80X3X8 
480  :  1920 
1  :  4,  Ans. 


20  : 

4  :  $  . 
%  0  :  $  4 


4 


1  :  4,  Ans. 


A  compound  ratio 
is  the  product  of  two 
or  more  simple  ra¬ 
tios  (Art.  358),  and 
the  product  of  two  or 
more  simple  ratios  is 
found  by  multiply¬ 
ing  the  antecedents  together  for  an  antecedent,  and  the  consequents 
for  a  consequent  (Art.  361).  Hence,  the  compound  ratio  given  is 
equal  to  20  X  4  X  6  :  80  X  3  X  8,  or  480  :  1920,  which,  by  dividing- 
both  terms  by  480,  is  reduced  to  1  :  4. 

The  process  may  be  shortened  by  canceling  the  factors  common  to 
the  product  of  the  antecedents  and  the  product  of  the  consequents. 

Note. — The  process  may  be  explained  directly  by  changing  each 
ratio  to  the  fractional  form,  thus : 


20  :  80  1 
4:3  \ 

6:8  J 

— .  20  V  4  V  6 

—  is  A  j  A  j 

?0Xi<X^  ,  , 

~$0X$X$  ' 

4  4 

Reduce  the  following  compound  ratios  to  simpl 

their  lowest  terms : 

• 

! 

f  6:8 

4.  J 

f  8:9 

2.  - 

10  :  12 

[  40:32 

1 

[  16  :  15 

1 

f  7:20 

1 

f  6:9 

5. 

1  40:21 

3.- 

15:18 

12  :  16 

1 

[  5:3 

1 

L  32  :  45 

f  4  .  5 1 

6.  Reduce  jg.’gf  :  :  16  :  15  to  a  simple  proportion. 

Suggestion. — Reduce  the  compound  ratio  to  a  simple  ratio. 


COMPOUND  PROPORTION. 


231 


Reduce  the  following  compound  proportions  to  simple 
proportions : 


7. 


f  8  :  12 
{21  :  10 


} 


42  :  30 


9.  $108  :  $216  : : 


f  24  :  36 

{  3:4 


8. 


(48  :  336  ) 

j  5  :  8  [ 

(.12:5  ) 


:  12  :  56 


f  6:9  4 
10.  \  12  :  3  [ 
(.15  :  36  3 


\l5  :  27 

1  12  :  6 


11.  What  is  the  fourth  term  of  the  compound  propor- 

(  5:8 ) 

tion,  -<  10  :  9  >■  : :  13  :  —  ? 

(  12  :  25J 


Process. 


0  :  M 

:  0  3  ::  13  :  — 

%!%:%%% _ 

1  :  3 : :  13  :  — 
_3 _ 

39,  Ans. 


Or: 

$ 

M 

tn 

03 

tn 

13 

3  X  13  =  39,  Ans. 


An  inspection  of  the  second  process  shows  that  the  four  numbers 
on  the  right  of  the  vertical  line  are  the  factors  of  the  product  of  the 
means,  and  that  the  three  numbers  on  the  left  are  the  factors  of  the  first 
extreme.  By  canceling  the  factors  common  to  dividend  and  divisor, 
the  fourth  term  is  found  directly. 


Find  the  fourth  term  of  these  compound  proportions: 


/* 


13.  { 


20  :  48' 
36  :  15 
10  :  4 

-  : :  25  :  — 

i4-l 

'5:9 
2.5  :  7.5 
4  :  10 

16  :  353 
21  :  8 

9  :  6 

12  :  45 

y  : :  16  :  — 

15.- 

f2i:7  3 

25  :  10 
4:6i 
(15  :  12  J 

376.  Rules. — 1.  To  reduce  a  compound  ratio  to  a  simple 
ratio,  Multiply  the  antecedents  together  for  an  antecedent ,  and 
the  consequents  for  a  consequent. 

2.  To  reduce  a  compound  proportion  to  a  simple  propor¬ 
tion,  Reduce  the  compound  ratio ,  or  each  compound  ratio ,  if 
there  are  two,  to  a  simple  ratio. 


232 


COMPLETE  ARITHMETIC. 


Case  II. 


The  Solution,  of  Problems  by  Compound 

Proportion.. 


16.  If  2  men  can  mow  16  acres  of  grass  in  10  days, 
working  8  hours  a  day,  how  many  men  can  mow  27  acres 
in  9  days,  working  10  hours  a  day  ? 


Statement. 


16  acres 
9  days 

10  hours 


27  acres 
10  days 
8  hours 


‘  U 

s  i 


2  men  :  Ans. 


2  :  3  : :  2  men  :  3  men,  Am. 


Or: 


Since  the  answer  required 
is  to  be  a  number  of  men, 
make  2  men  the  third  term. 

If  the  mowing  of  16  acres 
requires  2  men,  the  mowing 
of  27  acres  will  require  more 
than  2  men,  and  hence  the 
first  ratio  is  16  acres  :  27 
acres,  the  greater  number  be¬ 
ing  the  second  term. 

If  10  days  require  2  men, 
9  days  will  require  more  than 
2  men,  and  hence  the  second 
ratio  is  9  days  :  10  days,  the 
greater  number  being  the  sec¬ 
ond  term. 

If  working  8  hours  a  day  requires  2  men,  working  10  hours  a  day 
will  require  less  than  2  men,  and  hence  the  third  ratio  is  10  hours  :  8 
hours,  the  less  number  being  the  second  term. 

This  statement  gives  3  men  for  the  fourth  term. 


0 

10 

U 

$ 

% 

3 

Note. — In  determining  which  number  of  each  ratio  of  the  com¬ 
pound  ratio  is  to  be  the  second  term,  reason  from  the  number  in  the 
CONDITION. 


17.  If  12  men  can  build  50  rods  of  wall  in  15  days,  bow 
many  men  can  build  80  rods  in  16  days? 

18.  If  it  cost  $30  to  make  a  walk  10  feet  wide  and  90 
feet  long,  how  much  will  it  cost  to  make  a  walk  8  feet  wide 
and  225  feet  long? 

19.  If  6  men  can  excavate  576  cubic  feet  of  earth  in  8 
days  of  9  hours  each,  how  much  can  8  men  excavate  in  9 
days  of  10  hours  each? 


COMPOUND  PROPORTION. 


233 


20.  If  7  horses  eat  35  bushels  of  oats  in  25  days,  how 
many  bushels  will  15  horses  eat  in  21  days? 

21.  If  a  man  walk  120  miles  in  6  days  of  10  hours  each, 
how  many  miles  will  he  walk  in  16  days  of  8  hours  each  ? 

22.  If  1500  bricks,  each  8  in.  long  and  4  in.  wide,  will 
make  a  walk,  how  many  slabs  of  stone,  each  2  ft.  long  and 

1  ft.  4  in.  wide,  will  be  required  for  the  same  purpose? 

23.  If  the  interest  of  $250  for  9  months  is  $11.25,  what 
is  the  interest  of  $650  for  7  months? 

24.  If  it  cost  $84  to  carpet  a  room  36  ft.  long  and  21  ft. 
wide,  what  will  it  cost  to  carpet  a  room  33  ft.  long  and  27 
ft.  wide? 

25.  If  it  cost  $120  to  build  a  wall  40  ft.  long,  14  ft.  high, 
and  1  ft.  6  in.  thick,  what  will  it  cost  to  build  a  wall  180  ft. 
long,  21  ft.  high,  and  1  ft.  3  in.  thick? 

26.  If  4  men  can  dig  a  ditch  72  rd.  long,  5  ft.  wide,  and 

2  ft.  deep  in  12  days,  how  many  men  can  dig  a  ditch  120  rd. 
long,  6  ft.  wide,  and  1  ft.  6  in.  deep  in  9  days  ? 

27.  If  16  men  can  excavate  a  cellar  50  ft.  long,  36  ft. 
wide,  and  8  ft.  deep  in  10  days  of  8  hours  each,  in  how 
many  days  of  10  hours  each  can  6  men  excavate  a  cellar 
45  ft.  long,  25  ft.  wide,  and  6  ft.  deep? 

28.  If  32  men  can  dig  a  ditch  40  rd.  long,  6  ft.  wide, 
and  3  ft.  deep  in  9  days,  working  8  hours  a  day,  how  many 
men  can  dig  a  ditch  15  rd.  long,  4^  ft.  wide,  and  2  ft.  deep 
in  12  days,  working  6  hours  a  day? 

Note. — For  additional  problems,  see  Problems  for  Analysis. 

PRINCIPLES  AND  RULE. 

377.  Principles. — 1.  A  compound  proportion,  used  in  the 
solution  of  a  problem,  has  only  one  compound  ratio. 

2.  The  order  of  the  terms  of  each  ratio  composing  the  com¬ 
pound  ratio,  is  determined  as  in  simple  proportion. 

3.  The  fourth  term  of  a  compound  proportion  is  equal  to  the 
product  of  all  the  factors  of  the  second  and  third  terms,  divided 

by  the  product  of  the  factors  of  the  first  term. 

C.Ar.— 20. 


234 


COMPLETE  ARITHMETIC. 


378.  Rule. — 1.  Take  for  the  third  term  the  number  which 
is  of  the  same  kind  as  the  answer  sought,  and  arrange  the  first 
and  second  terms  of  each  ratio  composing  the  compound  ratio  as 
in  simple  proportion. 

2.  Reduce  the  compound  ratio  to  a  simple  ratio,  and  divide 
the  product  of  the  second  and  third  terms  of  the  resulting  pro¬ 
portion  by  the  first  term.  The  quotient  will  be  the  fourth  term, 
or  answer  sought.  Or, 

Divide  the  product  of  all  the  factors  of  the  second  and  third 
terms  of  the  compound  proportion  by  the  product  of  the  factors 
of  the  first  term,  shortening  the  process  by  cancellation. 

Notes. — 1.  The  terms  of  each  ratio  composing  the  compound  ratio 
are  arranged  precisely  as  they  would  be  if  the  answer  depended 
wholly  on  them  and  the  third  term. 

2.  The  process  of  solving  problems  by  compound  proportion  is 
also  called  “ The  Double  Rule  of  Three.” 


PARTNERSHIP. 


379.  A  Partnership  is  an  association  of  two  or  more 
persons  for  the  transaction  of  business. 

A  partnership  is  organized  and  regulated  by  a  contract,  called  arti¬ 
cles  of  agreement.  (Art.  240.) 

380.  A  partnership  association  is  called  a  Company,  Firm, 
or  House,  and  the  persons  associated  together  are  called 
Partners. 

381.  The  money  or  property  invested  in  the  business  by 
the  partners  is  called  Capital,  Joint-stock,  or  Stock  in  Trade. 

When  a  partner  furnishes  capital  but  does  not  assist  in  conducting 

the  business,  he  is  called  a  Silent  Partner. 

0 

382.  Partnership  is  either  Simple  or  Compound. 

In  Simple  Partnership*  the  capital  of  the  several 
partners  is  invested  an  equal  time. 

In  Compound  Partnership  the  capital  of  the  sev¬ 
eral  partners  is  invested  an  unequal  time. 


PARTNERSHIP. 


235 


SIMPLE  PARTNERSHIP. 


PROBLEMS. 

1.  A,  B,  and  C  entered  into  partnership  in  business  for 
2  years ;  A  put  in  $3600,  B  $2400,  and  C  $2000,  and  their 
net  profits  were  $3000.  What  was  each  partner’s  share  ? 


I.  Process  by  Proportion. 

$3600,  A’s  cap’l.  $8000  :  $3600  : :  $3000  :  $1350,  A’s  share  of  profits. 
2400,  B’s  cap’l.  $8000  :  $2400  : :  $3000  :  $900,  B’s  “  “ 

2000,  C’s  cap’l.  $8000  :  $2000  : :  $3000  :  $750,  C’s  “  “ 

$8000,  Entire  capital.  $3000,  Entire  profits. 

Since  the  capital  of  the  several  partners  was  employed  an  equal  time, 
their  shares  of  the  profits  are  proportional  to  their  capitals.  Hence, 
the  entire  capital  is  to  each  partner’s  capital  as  the  entire  profits  are 
to  his  share  of  the  profits. 


II.  Process  by 

$3000  -r-  $8000  =  .37£ 

$3600  X  -37£  =  $1350,  A’s  share. 
$2400  X  -37£  =  $900,  B’s  “ 
$2000  X  .37£  =  $750,  C’s  “ 

Or: 

$3600  -f-  $8000  =  .45,  A’i 
$2400  -f-  $8000  =  .30,  B’i 
$2000  -r-  $8000  =  .25,  C’i 
$3000  X  -45  =  $1350,  A’i 
$3000  X  -30  =  $900,  B’. 
$3000  X  -25  =  $750,  C’. 


Percentage. 

Since  the  profits  were  equal 
to  -37£,  or  37 \  %  of  the  entire 
capital,  each  partner’s  share  of 
the  profits  was  equal  to  37£  % 
of  his  capital. 

per  cent,  of  the  capital, 
u  u  « 

u  u  u 

share  of  the  profits, 
u  a  u 

u  <<  « 


III.  Process  by  Fractional  Parts. 


$3000  -f-  $8000  =  =  f. 

f  of  $3600  =  $1350,  A’s  share. 
|  of  $2400  =  $900,  B’s  “ 

|  of  $2000  =  $750,  C’s  “ 


Since  the  profits  were  equal  to 
f  of  the  entire  capital,  each  part¬ 
ner’s  share  of  the  profits  was  equal 
to  f  of  his  capital. 


236 


COMPLETE  ARITHMETIC. 


Or: 

$3600  -f-  $8000  =  29o>  A’s  part  of  the  capital. 

$2400  -T-  $8000  =  T3^,  B’s  “  “  “ 

$2000  =  $8000  =  C’s  “  “  “ 

-f-Q  of  $3000  =  $1350,  A’s  share  of  the  profits. 
x%  of  $3000  =$900,  B’s  “  “  “ 

4  of  $3000  =  $750,  C’s  “  “  “ 

Note. — Let  the  following  problems  be  solved  by  proportion  and  by 
either  of  the  other  methods,  which  the  teacher  or  pupil  may  prefer. 

2.  A  and  B  were  partners  in  business ;  A  put  in  $5000 
and  B  $4000,  and  their  profits  in  three  years  were  $4500: 
what  was  each  partner’s  share  of  the  profits? 

3.  A,  B,  and  C  formed  a  partnership  in  business;  A  put, 
in  $8000,  B  $4500,  and  C  $3500,  and  their  loss  the  first 
year  was  $3200 :  what  was  each  partner’s  share  ? 

4.  A,  B,  and  C  are  partners,  and  B  has  invested  }  as 
much  capital  as  A,  and  C  §  as  much  as  B :  if  their  profits 
amount  to  $6300,  what  will  be  each  partner’s  share  ? 

5.  The  capital  of  two  partners  is  proportional  to  4  and  3; 
their  profits  are  $10000  and  their  expenses  $2300:  what  is 
each  partner’s  share  of  the  net  profits? 

*6.  A,  B,  and  C  form  a  partnership,  A’s  capital  being 
$4000,  B’s  $6400,  and  C’s  $5600;  they  make  a  net  gain  of 
$3200,  and  then  sell  out  for  $20000:  what  is  each  partner’s 
share  of  the  gain  ?  Of  the  proceeds  of  the  sale  ? 

PRINCIPLES  AND  RULE. 

383.  Principles. — 1.  The  gain  or  loss  of  a  'partnership  is 
shared  by  the  partners  in  proportion  to  the  use  of  the  capital 
invested  by  them,  which  is  its  partnership  value. 

2.  When  the  time  is  equal,  the  use  of  the  capital  of  the  several 
partners  is  in  proportion  to  its  amount.  Hence, 

3.  In  a  simple  partnership,  the  gain  or  loss  is  shared  by  the 
partners  in  proportion  to  the  amounts  of  their  capital. 

384.  Bule. — To  divide  the  gain  or  loss  of  a  simple  part¬ 
nership,  Divide  the  gain  or  loss  among  the  several  partners  in 

proportion  to  the  amounts  of  capital  invested  by  them. 

*  Revised. 


COMPOUND  PARTNERSHIP. 


237 


Notes. — 1.  The  above  principles  and  rule  are  applicable  only  when 
the  several  partners  devote  equal  time  or  render  equal  service  in  car¬ 
rying  on  the  business.  The  division  of  profits  or  losses  is  usually 
settled  by  the  terms  of  the  contract. 

2.  The  problems  in  bankruptcy  (Art.  272)  may  also  be  solved  by 
the  above  methods. 


COMPOUND  PARTNERSHIP. 

7.  A  and  B  formed  a  partnership ;  A  put  in  $3000,  and, 
at  the  close  of  the  first  year,  added  $2000 ;  B  put  in  $4000, 
and,  at  the  close  of  the  second  year,  took  out  $2000 ;  at  the 
close  of  the  third  year,  the  profits  amounted  to  $3450. 
What  was  each  partner’s  share? 

I.  Process  by  Products. 

$3000  X  1=  $3000 

$5000  X  2  =  $10000 

$13000,  A’s  capital  for  1  year. 

$4000X  2=  $8000 
$2000X  1=  $2000 

$10000,  B’s  capital  for  1  year. 

$13000  -f-  $10000  =  $23000,  Entire  capital  for  1  year. 

$23000  :  $13000  : :  $3450  :  $1950,  A’s  share  of  profits. 

$23000  :  $10000  : :  $3450  :  $1500,  B’s  “  “ 

Since  A  had  $3000  invested  for  1  year  and  $5000  for  2  years,  the 
use  of  his  capital  was  equivalent  to  the  use  of  $13000  for  1  year. 
Since  B  had  $4000  invested  for  2  years  and  $2000  for  1  year,  the  use 

of  his  capital  was  equivalent  to  the  use  of  $10000  for  1  year.  Hence 

the  profits,  amounting  to  $3450,  should  be  shared  by  them  in  proportion 
to  $13000  and  $10000. 

II.  Process  by  Interest. 

Int.  of  $3000  for  1  yr.  =  $180 
“  “  $5000  for  2  yr.  =  $600 

$780,  Int.  of  A’s  capital. 

Int.  of  $4000  for  2  yr.  =  $480 
“  “  $2000  for  1  yr.  =  $120 

$600,  Int.  of  B’s  capital. 

$780  +  600  =  $1380,  Int.  of  entire  capital. 

$1380  :  $780  : :  $3450  :  $1950,  A’s  share. 

$1380  :  $600  : :  $3450  :  $1500,  B’s  “ 


238 


COMPLETE  ARITHMETIC. 


Since  the  use  of  capital  is  represented  by  its  interest  for  the  time, 
the  use  of  A’s  capital  is  represented  by  $780,  and  the  use  of  B’s  by 
$600.  Hence,  the  profits  ($3450)  should  be  shared  by  them  in  pro¬ 
portion  to  $780  and  $600. 

Note. — The  ratio  of  the  interests  will  be  the  same  whatever  be  the 
rate  per  cent ;  and  hence  the  interest  may  be  computed  at  any  rate. 

8.  A  and  B  entered  into  a  partnership  for  4  years, 
A  putting  in  $6000  and  B  $8000.  At  the  close  of  the 
second  year,  A  took  out  $2000  and  B  put  in  $2000 ;  and, 
at  the  close  of  the  fourth  year,  they  divided  $8890  as  net 
profits.  What  was  the  share  of  each? 

9.  A  and  B  entered  into  a  partnership  in  business  for  3 
years,  A’s  invested  capital  being  $3500  and  B’s  $4500.  At 
the  end  of  the  first  year  they  each  took  out  $1000,  and  C 
was  received  as  a  partner  with  a  capital  of  $2500.  At  the 
end  of  the  third  year  they  dissolved  partnership,  dividing 
$5000  as  net  profits.  What  was  each  partner’s  share? 

10.  A,  B,  and  C  entered  into  business  as  partners,  each 
putting  in  $5000  as  capital.  At  the  end  of  2  years  A  took 
out  $1000,  B  $2000,  and  C  $3000,  and,  at  the  end  of  the 
fourth  year,  they  closed  the  business  with  a  loss  of  $3600. 
What  was  the  loss  of  each? 

9 

PRINCIPLE  AND  RULES. 

385.  Principle. — The  value  of  capital  in  compound  partner¬ 
ship  depends  jointly  on  its  amount  and  the  time  of  its  investment. 

386.  Rules. — To  divide  the  gain  or  loss  of  a  compound 
partnership,  1.  Multiply  the  amount  of  capital  invested  by  each 
partner  by  the  time  of  its  investment,  and  taking  the  product  as 
the  partnership  value  of  his  capital,  proceed  as  in  simple  partner¬ 
ship.  Or, 

2.  Find  the  interest  of  each  partner's  capital  for  the  time  of  its 
investment,  at  any  rate  per  cent;  and  taking  the  interest  thus 
found  as  the  partnership  value  of  his  capital,  proceed  as  in  simple 
partnership. 


PROBLEMS  FOR  ANALYSIS. 


239 


PROBLEMS  FOR  ANALYSIS. 

Note. — These  problems  are  here  given  to  afford  an  additional 
drill  in  analysis  and,  if  needed,  in  proportion.  For  the  latter  pur¬ 
pose,  the  teacher  can  select  as  many  problems  as  may  be  necessary. 
Problems  marked  *  are  simplified  in  this  edition. 

MENTAL  PROBLEMS. 

1.  If  7  pounds  of  sugar  cost  91  cents,  what  will  20 
pounds  cost? 

2.  If  12  yards  of  muslin  cost  $1.02,  what  will  20  yards 
cost? 

3.  If  J  of  a  yard  of  silk  cost  what  will  £  of  a  yard 
cost? 

4.  If  J  of  a  barrel  of  flour  cost  $5£,  what  will  £  of  a 
barrel  cost? 

5.  If  £  of  a  pound  of  coffee  cost  15  cents,  what  will  3£ 
pounds  cost? 

6.  A  man  sold  a  watch  for  $120,  which  was  f  of  what 
it  cost  him :  how  much  did  it  cost? 

7.  If  40  yards  of  carpeting,  J  of  a  yard  wide,  will  cover 
a  floor,  how  many  yards  of  matting,  1£  yards  wide,  will 
cover  a  floor  of  equal  size? 

8.  Two  men,  traveling  in  the  same  direction,  are  60  miles 
apart;  the  one  in  advance  travels  5  miles  an  hour,  and  the 
other  7  miles  an  hour :  in  how  many  hours  will  the  latter 
overtake  the  former? 

9.  If  a  vertical  staff  3  feet  long  casts  a  shadow  2  feet 
in  length,  how  long  a  shadow  will  a  tree  90  feet  high  cast 
at  the  same  time  of  day? 

10.  If  a  steeple  200  feet  high  casts  a  shadow  150  feet 
long,  what  is  the  height  of  a  pole  which,  at  the  same  time 
of  day,  casts  a  shadow  80  feet  long? 

11.  If  5  men  can  do  a  piece  of  work  in  12  days,  how 
long  will  it  take  6  men  to  do  it? 

12.  If  8  men  can  do  a  piece  of  work  in  15  days,  how 
many  men  can  do  the  same  work  in  10  days? 


240 


COMPLETE  ARITHMETIC. 


13.  If  9  men  can  do  a  piece  of  work  in  4|  days,  how 
long  will  it  take  7  men  to  do  it? 

14.  If  3  pipes  will  empty  a  cistern  in  30  minutes,  how 
many  pipes  will  empty  it  in  10  minutes  ? 

15.  If  a  quantity  of  provisions  will  supply  15  men  20 
days,  how  long  will  it  supply  50  men? 

16.  If  it  require  12  days  of  10  hours  each  to  do  a  piece 
of  work,  how  many  days  of  8  hours  each  will  be  required 
to  do  the  same  work? 

17.  If  5  men  can  do  f  of  a  piece  of  work  in  a  day,  how 
long  will  it  take  one  man  to  do  the  entire  work? 

18.  If  8  men  can  do  j  of  a  piece  of  work  in  3  days,  how 
long  will  it  take  4  men  to  do  the  entire  work? 

19.  If  20  men  earn  $120  in  4  days,  how  much  will  5 
men  earn  in  8  days? 

20.  If  6  men  can  mow  30  acres  of  grass  in  3  days,  how 
many  acres  will  9  men  mow  in  5  days? 

21.  If  5  horses  eat  40  bushels  of  oats  in  3  weeks,  how 
many  bushels  will  supply  12  horses  10  weeks? 

22.  If  8  men  can  dig  a  ditch  40  rods  long  in  6  days,  how 
long  will  it  take  12  men  to  dig  a  ditch  60  rods  long? 

23.  If  the  interest  of  $50  for  9  months  is  $6,  what  would 
be  the  interest  of  $150  for  1  yr.  6  mo.? 

24.  A  school  enrolls  180  pupils,  and  the  number  of  boys 
is  f  of  the  number  of  girls:  how  many  pupils  of  each  sex 
are  enrolled  in  the  school  ? 

25.  A  lady  paid  $130  for  a  watch  and  chain,  and  the  cost 
of  the  watch  was  f  more  than  the  cost  of  the  chain  :  what 
was  the  cost  of  each  ? 

26.  A  tree  120  feet  in  height  was  broken  into  two  parts 
by  falling,  and  §  of  the  shorter  part  equaled  f-  of  the 
longer :  what  was  the  length  of  each  part  ? 

27.  A  person  giving  the  time  of  day,  said  that  •§  of  the 
time  past  noon  equaled  the  time  to  midnight :  what  was  the 
hour  of  day?  [Sug.:  §  t.  past  n.  -f-  J  t.  past  n.  =  12  h.] 

28.  A  person  being  asked  the  time  of  day,  said  that  J  of 
the  time  past  midnight  equaled  the  time  to  noon :  what  was 
the  hour  of  day  ? 


PEOBLEMS  FOE  ANALYSIS. 


241 


29.  What  is  the  hour  of  day  when  f  of  the  time  past 
noon  equals  f  of  the  time  to  midnight? 

Suggestion:  If  f  t.  past  n.  =  §  t.  to  m.,  t.  past  n.  =  4  t.  to  m. 
Hence,  t.  to  m.-f-i  t.  to  m.  =  12  h. 

*30.  What  is  the  time  of  day  when  §  of  the  time  past 
noon  is  equal  to  twice  the  time  to  midnight? 

31.  What  is  the  time  of  day  when  §  of  the  time  to  noon 
is  equal  to  J-  of  the  time  past  midnight? 

32.  A  man  being  asked  his  age  said,  10  years  ago  my 
age  was  -J  of  my  present  age:  what  was  his  age? 

33.  A  son’s  age  is  f  of  the  age  of  his  father,  and  the 
sum  of  their  ages  is  80  years:  what  is  the  age  of  each? 

*34.  Ten  years  ago  A’s  age  was  f  of  B’s  age,  and  the  sum 
of  their  ages  was  70  years:  what  is  the  present  age  of  each? 

*35.  At  the  time  of  marriage  a  wife’s  age  was  f  of  the 
age  of  her  husband,  and  the  sum  of  their  ages  was  48  years : 
how  old  wras  each  20  years  after  marriage? 

36.  f  of  A’s  age  equals  of  B’s,  and  the  difference  be¬ 
tween  their  ages  is  10  years:  how  old  is  each? 

37.  Twice  the  age  of  A  is  20  years  more  than  the  age 
of  B,  and  10  years  more  than  the  age  of  C,  and  the  sum 
of  their  ages  is  120  years:  what  is  the  age  of  each? 

*38.  A  man  bought  a  horse  and  carriage  for  $275,  and 
J  of  the  cost  of  the  horse  equals  J  of  the  cost  of  the  car¬ 
riage  :  what  was  the  cost  of  each  ? 

39.  A  man  bought  a  horse,  saddle,  and  bridle  for  $150; 
the  cost  of  the  saddle  was  of  the  cost  of  the  horse,  and 
the  cost  of  the  bridle  was  of  the  cost  of  the  saddle :  what 
was  the  cost  of  each? 

40.  A  man  and  his  two  sons  earned  $140  a  month ;  the 
man  earned  twice  as  much  as  the  elder  son,  and  the  elder 
son  twice  as  much  as  the  younger :  how  much  did  each  earn? 

41.  Two  men  bought  a  barrel  of  syrup,  one  paying  $20 
and  the  other  $30 :  what  part  should  each  have  ? 

42.  Two  men  hired  a  pasture  for  $40,  and  one  put  in  3 

cows  and  the  other  5  cows:  how  much  ought  each  to  pay? 

C.  Ar.— 21. 


242 


COMPLETE  ARITHMETIC. 


43.  A  and  B  rented  a  pasture  for  $72 ;  A  puts  in  40 
sheep  and  B  8  cows :  if  4  sheep  eat  as  much  as  one  cow, 
how  much  ought  each  to  pay? 

44.  Two  men,  A  and  B,  agreed  to  build  a  wall  for  $300; 
A  sent  5  men  for  4  days,  and  B  5  men  for  6  days  :  how 
much  ought  each  to  receive  ? 

45.  A  and  B  engage  to  plow  a  field  for  $81 ;  A  furnished 
3  teams  for  5  days,  and  B  furnished  4  teams  for  3  days  : 
how  much  should  each  receive  ? 

46.  A  man  can  do  \  of  a  piece  of  work  in  a  day,  and  a 
boy  can  do  of  it  in  a  day:  in  how  many  days  can  both 
of  them,  working  together,  do  it  ? 

47.  A  and  B  together  can  build  a  wall  in  8  days,  and  A 

can  build  it  alone  in  12  days :  how  long  will  it  take  B  to 

build  it? 

48.  A  can  do  a  piece  of  work  in  6  days,  and  B  in  8 

days  :  if  they  both  work  together  3  days,  how  long  will  it 

take  B  alone  to  complete  the  work  ? 

49.  John  can  saw  a  pile  of  wood  in  6  days,  and,  with  the 
assistance  of  Charles,  he  can  saw  it  in  4  days :  how  long 
will  it  take  Charles  to  saw  it  alone  ? 

*50.  A  man  can  do  a  piece  of  work  in  4  days  and  a  boy 
in  8  days  ;  the  man  works  2  days  alone  and  is  then  assisted 
by  the  boy:  how  long  will  it  take  both  to  complete  the 
work  ? 

51.  A  and  B  can  do  a  piece  of  work  in  10  days,  and  A, 
B,  and  C  in  8  days :  how  long  will  it  take  C  alone  to  do 
the  work? 

52.  A  and  B  can  do  -J-  of  a  piece  of  work  in  a  day,  and 
A  can  do  twice  as  much  in  a  day  as  B :  how  long  will  it 
take  B  alone  to  do  it? 

53.  A  can  make  a  fence  in  ^  of  a  month,  B  in  ^  of  a 
month,  and  C  in  jr  of  a  month :  in  what  time  can  all  three 
together  build  it? 

54.  A  can  do  a  piece  of  work  in  4  days,  B  in  5  days, 
and  C  in  6  days :  in  what  time  can  they  together  do  it? 
*55.  A,  B,  and  C  can  do  a  piece  of  work  in  4  days,  A 


PROBLEMS  FOR  ANALYSIS. 


243 


and  C  in  8  days,  and  B  and  C  in  6  days:  how  long  will 
it  take  each,  working  alone,  to  do  it? 

56.  A  and  B  did  a  piece  of  work,  and  f  of  what  A  did 
equaled  f  of  what  B  did :  if  B  received  $18,  how  much  did 
A  receive? 

57.  A  man  spent  f  of  his  money,  and  then  earned  \  as 
much  as  he  had  spent,  and  then  had  $21  less  than  he  had 
at  first:  how  much  money  did  he  have  at  first? 

58.  At  what  time  between  one  and  two  o’clock  will  the 
hour  and  minute  hands  of  a  watch  be  together? 

Note. — For  solution,  see  page  286. 

59.  At  what  time  between  two  and  three  o’clock  are  the 
hour  and  minute  hands  of  a  watch  together?  At  what 
time  between  four  and  five  o’clock? 

*60.  A  man  bought  two  watches  and  a  chain  for  $140; 
and  the  first  watch  cost  twice  as  much  as  the  chain,  and  the 
second  watch  four  times  as  much  as  the  chain:  what  was 
the  cost  of  each? 

WRITTEN  PROBLEMS. 

61.  A  father  bequeathed  $14535  to  two  sons,  giving  the 
younger  ^  as  much  as  the  elder:  what  was  the  share  of 
each  ? 

62.  An  estate  was  so  divided  between  two  heirs  that  §  of 
the  share  of  the  elder  was  equal  to  f  of  the  share  of  the 
younger,  and  the  difference  between  their  shares  was  $362 : 
what  was  the  share  of  each? 

63.  An  estate  worth  $27520  was  divided  between  two 
daughters  in  proportion  to  their  ages,  which  were  14  and 
18  years  respectively :  how  much  did  each  receive  ? 

64.  A  man  paid  $8100  for  2  farms,  and  f  of  the  cost  of 
the  larger  farm  was  equal  to  ^  of  the  cost  of  the  smaller: 
what  was  the  cost  of  each  ? 

65.  A  earns  $15.50  as  often  as  B  earns  $12.40,  and  in  a 
certain  time  they  together  earn  $697.50 :  how  much  did 
each  earn? 

66.  The  fore  wheels  of  a  carriage  arc  each  9^  feet  in  cir- 


244 


COMPLETE  ARITHMETIC. 


cumference,  and  the  hind  wheels  are  each  12^-  feet  in  cir¬ 
cumference  :  if  each  fore  wheel  revolve  9500  times  in  going 
a  certain  distance,  how  many  times  will  each  hind  wheel 
revolve  ? 

67.  If  it  take  13200  steps  of  2  ft.  9  in.  each  to  walk  a 
certain  distance,  how  many  steps  of  1  ft.  10  in.  each  will 
it  take  to  walk  the  same  distance  ? 

68.  If  $75  yield  $10.80  interest,  what  principal  will  yield 
$89.28  interest  in  the  same  time? 

69.  If  the  interest  of  $475  is  $118.75,  what  would  be  the 
interest  of  $850  for  the  same  time  and  at  the  same  rate  ? 

70.  If  the  interest  of  a  certain  principal  for  a  certain 
time  at  5  per  cent  is  $120.50,  what  would  he  the  interest 
of  the  same  principal  for  the  same  time  at  12  per  cent? 

71.  A  broker  sold  90  shares  of  railroad  stock  and  gained 
$315:  how  much  would  he  have  gained  if  he  had  sold  245 
shares  ? 

72.  If  a  gain  of  15  per  cent  on  a  certain  investment 
yields  $2347.50,  what  would  a  gain  of  24  per  cent  on  the 
same  investment  yield  ? 

73.  If  the  commission  for  selling  3050  pounds  of  butter 
at  30  cents  a  pound  is  $45.75,  what  would  be  the  com¬ 
mission  for  selling  7500  pounds  at  35  cents  a  pound? 

74.  If  the  annual  dividend  on  $40325  worth  of  mining 
stock  is  $3226,  what  is  the  dividend  on  $70680  of  the  same 
stock  ? 

75.  If  6  ranks  of  wood,  each  60  ft.  long  and  6  ft.  high, 
are  worth  $337.50,  what  is  the  value  of  15  ranks  of  wood, 
each  45  ft.  long  and  9  ft.  high? 

76.  If  it  cost  $110  to  dig  a  cellar  40  ft.  long,  27  ft.  wide, 
and  4  ft.  deep,  how  much  will  it  cost  to  dig  a  cellar  36  ft. 
long,  30  ft.  wide,  and  5  ft.  deep? 

77.  If  45  men  can  do  a  piece  of  work  in  15  days,  by 
working  8  hours  a  day,  in  how  many  days  can  30  men, 
Working  9  hours  a  day,  do  the  same  work? 

78.  If  5  men  can  cut  45  cords  of  wood  in  6  days,  how 
many  cords  can  8  men  cut  in  15  days? 


PROBLEMS  FOR  ANALYSIS. 


245 


79.  If  4  men  dig  a  trench  in  15  days  of  10  hours  each, 
in  how  many  days  of  8  hours  each  can  5  men  perform  the 
same  work? 

80.  A  and  B  are  partners  in  business ;  A’s  capital  is 
equal  to  f  of  B’s,  and  their  profits  are  $3250 :  what  is  the 
share  of  each? 

81.  A  and  B  are  partners;  f  of  A’s  capital  is  equal  to  f 
of  B’s,  and  their  loss  in  business  is  $2150:  what  is  the 
share  of  each? 

82.  A,  B,  and  C  are  partners  in  business ;  A’s  capital  is 
twice  B’s  and  three  times  C’s,  and  their  profits  in  business 
are  $4675:  what  is  the  share  of  each? 

83.  A  and  B,  trading  in  partnership  2  years,  make  a 
profit  of  $5460 ;  during  the  first  year  A  owned  |-  of  the 
stock,  and  during  the  second  year  B  owned  f  of  it:  what 
is  each  partner’s  share  of  the  profits? 

84.  A  and  B,  trading  in  partnership  2  years,  make  each 
year  a  profit  of  $1200 ;  A’s  capital  the  first  year  was  2^  times 
B’s,  and  the  second  year  it  was  1 J  times  B’s :  what  is  each 
partner’s  share  of  the  profits  ? 

85.  A  and  B  traded  in  partnership  3  years;  A’s  stock 
the  first  year  was  $5000,  the  second  year  $6000,  and  the 
third  year  $7000;  B’s  stock  the  first  year  was  $7000,  and 
the  last  two  years  $5000;  their  loss  was  $1750.  What  was 
the  loss  of  each  ? 

86.  A  mechanic  agreed  to  work  80  days  on  the  condition 
that  he  should  receive  $1.75  and  board  for  every  day  that 
he  worked,  and  that  he  should  pay  75  cents  a  day  for  board 
when  he  was  idle ;  his  net  earnings  for  the  time  were  $80 : 
how  many  days  did  he  work? 

Sug. — Amount  of  loss-v-$2.50  (loss  each  idle  day)  =  no.  idle  days. 

87.  A  piece  of  carpeting  containing  135  yards  was  cut 
into  3  carpets,  and  of  the  number  of  yards  in  the  first 
carpet  was  equal  to  ^  of  the  number  of  yards  in  the  second 
carpet,  and  to  §  of  the  number  of  yards  in  the  third  carpet: 
what  was  the  number  of  yards  in  each  carpet? 


246 


COMPLETE  ARITHMETIC. 


SECTION  XVI. 

INVOLUTION  AND  EVOLUTION. 

I.  INVOLUTION. 

387.  The  first  power  of  4  is  4;  the  second  power  of  4  is 
4x4,  which  is  16;  the  third  power  is  4  X  4  X  4,  which  is 
64 ;  the  fourth  power  is  4  X  4  X  4  X  4,  which  is  256  ;  etc. 

1.  What  is  the  second  power  of  5?  Of  6?  8?  10? 

2.  What  is  the  third  power  of  3 ?  Of  4?  5?  6?  10? 

3.  What  is  the  fourth  power  of  2  ?  Of  3  ?  4  ?  10  ? 

4.  What  is  the  second  power  ofl?  2?  3?  4?  5? 
6?  7?  8?  9? 

5.  What  is  the  third  power  ofl?  2?  3?  4?  5?  6? 
7?  8?  9? 

6.  What  is  the  second  power  of  f?  Off?  £?  f? 

i  ?  A  ? 

6  *  6  ' 

WRITTEN  PROBLEMS. 

7.  What  is  the  second  power  of  406? 

8.  What  is  the  third  power  of  42  ? 

9.  What  is  the  fourth  power  of  24? 

10.  What  is  the  fifth  power  of  16? 

11.  What  is  the  second  power  of  6.5? 

12.  What  is  the  third  power  of  .42? 

13.  What  is  the  fifth  power  of  .6? 

14.  What  is  the  third  power  of  ^-f. 

15.  What  is  the  second  power  of  ? 

16.  What  is  the  fourth  power  of  f-J-? 

Remark. — The  power  to  which  a  number  is  to  be  raised  may  be 
denoted  by  a  little  figure,  called  an  exponent ,  placed  at  the  right  of 
the  upper  part  of  the  figures  expressing  the  number.  Thus,  24 2 
denotes  the  second  power  of  24;  16 3  denotes  the  third  power  of  16, 
etc. 


INVOLUTION. 


247 


Raise  the  following  numbers  to  the  powers  indicated  by 
the  exponents : 


17.  623 2 

18.  105 3 

19.  34. 6 2 

20.  .0163 

21.  1.44 


22.  .045 2 

23.  (If)3 

24.  (J)4 

25.  (16f)2 

26.  (3i) 4 


27.  (6J) 8 

28.  (if)2 

29.  .005 5 

30.  2.043 

31.  (|)4 


DEFINITIONS  AND  RULE. 

388.  The  Fower  of  a  number  is  the  product  obtained 
by  taking  the  number  one  or  more  times  as  a  factor. 

389.  The  First  Fower  of  a  number  is  the  number 
itself. 

390.  The  Second  Fower  of  a  number  is  the  product 
obtained  by  taking  the  number  twice  as 
a  factor. 

It  is  also  called  the  Square  of  the  number, 
since  the  area  of  a  geometrical  square  is  rep¬ 
resented  by  the  product  obtained  by  taking 
the  number  of  linear  units  in  one  of  its  sides 
twice  as  a  factor. 


391.  The  Third  Fower  of  a  number  is  the  product 
obtained  by  taking  the  number  three 
times  as  a  factor. 

It  is  also  called  the  Cube  of  the  number, 
since  the  capacity  of  a  geometrical  cube  is 
represented  by  the  product  obtained  by  tak¬ 
ing  the  number  of  linear  units  in  one  of  its 
edges  three  times  as  a  factor. 

3  x  3  x  3  =  27. 

392.  The  Exponent  of  a  power  is  a  small  figure  placed  at 
the  right  of  the  number,  to  show  how  many  times  it  is  to 
be  taken  as  a  factor.  It  denotes  the  degree  of  the  power. 


248 


COMPLETE  ARITHMETIC. 


The  first  power  contains  the  number  once  as  a  factor,  and  the  expo¬ 
nent  is *  1 ;  the  second  power,  or  square,  contains  the  number  twice  as 
a  factor,  and  the  exponent  is  2  ;  the  third  power,  or  cube,  contains 
the  number  three  times  as  a  factor,  and  the  exponent  is  3  ;  etc. 

393.  Involution  is  the  process  of  finding  the  powers 
of  numbers. 

394.  Rule. — To  raise  a  number  to  a  given  power,  Mul¬ 
tiply  the  number  by  itself  as  many  times  less  one  as  there  are 
units  in  the  exponent  of  the  given  power.  The  last  product  will 
be  the  required  power. 


395.  ANOTHER  METHOD  OF  INVOLUTION. 


32.  What  is  the  square  of  53  ? 

Process. 

53  =  50  +  3,  and  532  =  (50  +  3)2 
50  +  3 
50  +  3 


50X3  +  32  =  (50  +  3)  X  3 
50 2  +  50  X  3  =  (50  +  3)  X  50 


Parts  added. 
50 2  =  2500 
2(50X3)=  300 

32  = _ 9 

=  2809 


50 2  +  2  (50  X  3)  +  32  =  (50  +  3)2  =  532 

An  inspection  of  the  above  process  will  show  that  the  square  of  53 
is  equal  to  the  square  of  the  5  tens,  plus  twice  the  product  of  the  5 
tens  by  the  3  units,  plus  the  square  of  the  units. 

In  like  manner,  it  may  be  shown  that  the  square  of  any  number, 
composed  of  tens  and  units,  is  equal  to  The  square  of  the  tens ,  plus  twice 
the  product  of  the  tens  by  the  units ,  plus  the  square  of  the  units. 

33.  What  is  the  square  of  45? 

(  402  =1600 

Process:  452  =  j  ^(40X5)=  400 

I  2025,  Ans. 

34.  What  is  the  square  of  67?  Of  75? 

35.  What  is  the  square  of  82  ?  Of  38  ? 

36.  What  is  the  square  of  93  ?  Of  125  ? 

Suggestion. — 125  =  120  +  5. 

37.  What  is  the  square  of  115?  Of  124? 


EVOLUTION. 


249 


38.  What  is  the  cube  of  53? 


The  cube  of  53  =  (50  +  3) 8  =  503  +  3  (502X  3)  +  3  (50  X  32)  -f  33, 
as  may  be  shown  by  multiplying  50 2  +  2  (50  X  3)  +  3 2  by  50  -}-  3. 

In  like  manner,  it  may  be  shown  that  the  cube  of  any  number, 
composed  of  tens  and  units,  is  equal  to  The  cube  of  the  tens,  plus  three 
times  the  product  of  the  square  of  the  tens  by  the  units,  plus  three  times  the 
product  of  the  tens  by  the  square  of  the  units,  plus  the  cube  of  the  units. 


39.  What  is  the  cube  of  45  ? 


Process:  45 3  = 


(  40 3  =  64000 

3  (402  X  5)  =  24000 
3  (40  X  52)  =  3000 
53  =  125 

^  91125,  Ans. 


40.  What  is  the  cube  of  23?  Of  32? 

41.  What  is  the  cube  of  24?  Of  43? 

42.  What  is  the  cube  of  33  ?  Of  54  ? 

43.  What  is  the  cube  of  51  ?  Of  35  ? 

44.  What  is  the  cube  of  45?  Of  52? 

45.  What  is  the  cube  of  41  ?  55? 

46.  What  is  the  cube  of  75?  80? 


II.  EVOLUTION. 

MENTAL  PROBLEMS. 

1.  What  are  the  two  equal  factors  of  16?  Of  25?  49? 

2.  Of  what  number  is  81  the  second  power  or  square  ? 

3.  What  are  the  three  equal  factors  of  8  ?  Of  27  ?  125  ? 

4.  Of  what  number  is  125  the  third  power  or  cube  ? 

One  of  the  two  equal  factors  of  a  number  is  called  its  second  or 
square  root;  one  of  its  three  equal  factors,  its  third  or  cube  root;  one 
of  its  four  equal  factors,  its  fourth  root ,  etc. 

5  What  is  the  square  root  of  25  ?  Of  49  ?  64?  81  ? 

6.  What  is  the  cube  root  of  8  ?  Of  27  ?  64  ?  125  ? 

7.  What  is  the  cube  root  of  216?  512?  1000? 

8.  What  is  the  fourth  root  of  16?  Of  81?  256?  625? 

9.  What  is  the  square  root  of  1  ?  4  ?  9  ?  16  ?  25  ? 

36?  49?  64?  81? 


250 


COMPLETE  ARITHMETIC. 


10.  What  is  the  cube  root  of  1  ?  8  ?  27  ?  64  ?  125  ? 
216  ?  343  ?  512  ?  729  ? 

11.  What  integers  between  1  and  100  are  perfect  squares? 

12.  What  integers  between  1  and  1000  are  perfect  cubes  ? 

13.  Show  that  the  square  root  of  a  perfect  square  ex¬ 
pressed  by  two  figures,  can  not  exceed  9. 

14.  Show  that  the  cube  root  of  a  perfect  cube  expressed 
by  three  figures,  can  not  exceed  9. 

DEFINITIONS. 

396.  The  Hoot  of  a  number  is  one  of  the  equal  factors 
which  will  produce  it. 

397.  The  First  Hoot  is  the  number  itself. 

398.  The  Second  Hoot  is  one  of  the  two  equal  factors 
of  the  number.  It  is  also  called  the  Square  Root. 

399.  The  Third  Hoot  is  one  of  the  three  equal  factors 
of  the  number.  It  is  also  called  the  Cube  Root 

A  number  is  the  second  power  of  its  square  root;  the  third  power 
of  its  cube  root ;  the  fourth  power  of  its  fourth  root ;  etc. 

400.  A  Ferfect  Fower  is  the  product  of  equal  factors. 
It  has  an  exact  root. 

401.  An  Imperfect  Fower  is  a  number  which  is  not 
the  product  of  equal  factors.  Its  root  is  called  a  Surd. 

402.  The  Hcidical  Sign  is  a  character,  y"  ,  placed 
before  a  number  to  show  that  its  root  is  to  be  taken. 

403.  A  small  figure  placed  above  the  radical  sign  is 
called  the  Index  of  the  root. 

Thus,  1^25  denotes  the  first  root  of  25  ;  ^ 25  denotes  the 
second  root  of  25  ;  1^25,  the  third  root  of  25,  etc. 


SQUARE  ROOT. 


251 


When  the  square  root  is  indicated,  the  index  is  usually  omitted. 
V 16  and  V 16  alike  denote  the  square  root  of  16. 

Note. — The  root  of  a  number  may  also  be  indicated  by  a  frac¬ 
tional  exponent.  Thus,  162  denotes  the  square  root  of  16 ;  16^,  the 
cube  root  of  16,  etc. ;  16^  denotes  the  cube  root  of  the  square  of  16. 

404.  J Evolution  is  the  process  of  finding  the  roots  of 
numbers. 

Note. — Evolution  is  the  reverse  of  involution. 


SQUARE  ROOT. 


The  Division  of  the  IN- umber  into  -Periods. 


405.  The  smallest  integer  composed  of  one  order  of  fig¬ 
ures  is  1,  and  the  greatest  is  9 ;  the  smallest  integer  com¬ 
posed  of  two  orders  is  10,  and  the  greatest  is  99,  and  so  on. 

The  squares  of  the  smallest  and  the  greatest  integers  com¬ 
posed  of  one,  two,  three,  and  four  orders,  are  as  follows : 


12  =  1 

102  =  100 

1002  =  10000 

1000 2 = 1000000 


92  =  81 

99 2  =  9801 

9992  =  998001 

99992  = 99980001 


A  comparison  of  the  above  numbers  with  their  squares 
shows  that  the  square  of  a  number  contains  twice  as  many 
orders  as  the  number,  or  twice  as  many  orders  less  one. 

406.  Hence,  if  a  number  be  separated  into  periods  of 
two  orders  each,  beginning  at  the  right,  there  will  be  as  many 
orders  in  its  square  root  as  there  are  periods  in  the  number . 

1.  How  many  orders  in  the  square  root  of  2809? 

Suggestion. — First  divide  the  number  into  periods  of  two  orders 
each,  thus:  2809. 

2.  How  many  orders  in  the  square  root  of  36864? 

3.  How  many  orders  in  the  square  root  of  345744? 


252 


COMPLETE  ARITHMETIC. 


4.  How  many  orders  in  the  square  root  of  87616? 

5.  How  many  orders  in  the  square  root  of  5308416  ? 

6.  How  many  orders  in  the  square  root  of  5475600? 

7.  How  many  orders  in  the  square  root  of  14440000? 

407.  The  squares  of  the  smallest  and  the  greatest  number 
of  units,  tens,  hundreds,  and  thousands,  are  as  follows : 


l2  =  1 

102  =  100 
1002  =  10000 
10002 = 1000000 


92  =  81 

902  =  8100 

9002  =  810000 

9000 2  =81000000 


A  comparison  of  the  above  numbers  with  their  squares 
shows  that  the  square  of  units  gives  no  order  higher  than 
tens ;  that  the  square  of  tens  gives  no  order  lower  than 
hundreds,  nor  higher  than  thousands ;  that  the  square  of 
hundreds  gives  no  order  lower  than  ten-thousands,  nor 
higher  than  hundred-thousands,  etc. 

408.  Hence,  if  a  number  be  separated  into  periods  of  two 
orders  each,  the  left-hand  period  will  contain  the  square  of  the 
left-hand  or  first  term  of  the  square  root ;  the  first  two  left-hand 
periods  will  contain  the  square  of  the  first  two  terms  of  the  square 
root,  etc. 

8.  What  is  the  tens’  term  of  the  square  root  of  2025  ? 

Ans. — The  left-hand  period  of  2025  is  20 ;  the  greatest  square  in 
20  is  16,  and  the  square  root  of  16  is  4.  Hence,  the  tens’  figure  of 
the  square  root  of  2025  is  4. 

9.  What  is  the  hundreds’  term  of  the  square  root  of' 
87616?  Of  345741  ? 

10.  What  is  the  left-hand  term  of  the  square  root  of 
16129?  Of  336400? 

11.  What  is  the  left-hand  term  of  the  square  root  of 
87616? 

12.  What  are  the  first  two  terms  of  the  square  root  of 
16129? 


SQUARE  ROOT. 


253 


WRITTEN  PROBLEMS. 


13.  What  is  the  square  root  of  3364? 


Process. 

3364  I  58 
52  =_25 

5X2  =  10)  864 
108X8=  864 


Since  3364  is  composed  of  two  periods, 
its  square  root  will  be  composed  of  two 
orders.  (Art.  406.) 

The  left-hand  period  33  contains  the 
square  of  the  tens’  term  of  the  root.  (Art. 
408.)  The  greatest  square  in  33  is  25,  and 
the  square  of  25  is  5.  Hence,  5  is  the  tens’  term  of  the  root. 

The  square  of  a  number  composed  of  tens  and  units  is  equal  to 
the  square  of  the  tens  plus  twice  the  product  of  the  tens  by  the  units, 
plus  the  square  of  the  units.  (Art.  395).  Hence,  the  difference  be¬ 
tween  3364  and  the  square  of  the  5  tens  of  its  root,  is  composed  of 
twice  the  product  of  the  tens  of  the  root  by  the  units ,  plus  the  square  of  the 
units. 

But  the  product  of  tens  by  units  contain  no  order  lower  than  tens, 
and  hence  the  86  tens  in  the  864,  the  difference,  contains  twice  the 
product  of  the  tens  by  the  units.  Hence,  if  the  86  tens  be  divided  by 
twice  the  5  tens  of  the  root,  the  quotient,  which  is  8,  will  be  the  units’ 
term  of  the  root. 

If  the  8  units  be  annexed  to  the  10  tens,  used  as  a  trial  divisor,  and 
the  result,  108,  be  multiplied  by  8,  the  product  will  be  twice  the  prod¬ 
uct  of  the  tens  of  the  root  by  the  units,  plus  the  square  of  the  units. 
108X  8=2  (5X  8)  +82. 

Proof. —  58  X  58  =  3364. 


14.  What 

15.  What 

16.  What 

17.  What 

18.  What 
133225? 

19.  What 
210681 ? 

20.  What 
419904  ? 

21.  What 
94249  ?  Of 


is  the  square  root  of  625?  Of  4225? 
is  the  square  root  of  576?  Of  7744? 
is  the  square  root  of  1444?  Of  6241? 
is  the  square  root  of  3025?  Of  7569? 
is  the  square  root  of 


is  the  square  root  of 
is  the  square  root  of 


Process. 


133225  I  365 
9 

3  X  2  =  6  )  43  2 
66  X  6  =  396 
36X  2  =  72)  362  5 
is  the  square  root  of  725  X  5  =  3625 

492804  ? 


254 


COMPLETE  ARITHMETIC. 


22.  What  is  the  square  root  of  57600?  Of  40960000? 

23.  What  is  the  square  root  of 
10.4976? 

24.  What  is  the  square  root  of 
176.89? 

25.  What  is  the  square  root  of 
.0625? 

26.  What  is  the  square  root  of 
.451584?  Of  .008836? 

27.  What  is  the  square  root  of  586.7? 

Suggestion. — Point  thus  586.70,  and  carry  the  root  to  three  deci¬ 
mal  places  by  annexing  periods  of  decimal  ciphers. 

28.  What  is  the  square  root  of  75.364?  Of  5.493? 

29.  What  is  the  square  root  of  263.85?  Of  13467? 

30.  What  is  the  square  root  of  -§-§-J?  Of 

31.  What  is  the  square  root  of  272^?  Of  1040^? 

32.  What  is  the  square  root  of  2  ?  Of  3  ?  Of  5  ? 

PRINCIPLES  AND  RULE. 

409.  Principles. — 1.  The  square  root  of  a  number  contains 
as  many  orders  as  there  are  periods  of  two  orders  each  in  the 
number . 

2.  The  left-hand  period  of  a  number  contains  the  square  of 
the  first  term  of  its  square  root. 

3.  The  square  of  a  number ,  composed  of  tens  and  units ,  is 
equal  to  the  square  of  the  tens,  plus  twice  the  product  of  the  tens 
by  the  units,  plus  the  square  of  the  units. 

410.  Rule. — To  extract  the  square  root  of  a  number, 

1.  Begin  at  the  units’  order  and  separate  the  number  into 
periods  of  two  orders  each. 

2.  Find  the  greatest  perfect  square  in  the  left-hand  period, 
and  place  its  square  root  at  the  right  for  the  first  or  highest  term 
of  the  root. 

3.  Subtract  the  square  of  the  term  of  the  root  found  from  the 


Process. 

10.4976  |  3.24 
9 

3X2  =  6)1.49 
6.2  X  2  =  1.24 
3.2X2=  6.4)  .257  6 
6.44  X  .04  =  .2576 


SQUARE  ROOT. 


255 


left-hand  period ,  and  to  the  difference  annex  the  second  period 
for  a  dividend. 


4.  Take  twice  the  term  of  the  root  found  for  a  trial  divisor, 
and  the  dividend,  exclusive  of  its  right-hand  figure,  for  a  trial 
dividend.  The  quotient  (or  the  quotient  reduced )  will  be  the 
next  term  of  the  root. 


5.  Annex  the  second  term  of  the  root  to  the  trial  divisor,  and 
multiply  the  result  by  the  second  term,  and  subtract  the  product 
from  the  dividend. 


6.  Annex  the  third  period  to  the  remainder  for  the  next  divi¬ 
dend,  and  divide  the  same,  exclusive  of  the  right-hand  figure,  by 
twice  the  terms  of  the  root  found;  and  continue  in  like  manner 


until  all  the  periods  are  used. 

Notes. — 1.  The  left-hand  period  may  contain  but  one  order. 

2.  Twice  the  term  or  terms  of  the  root,  as  the  case  may  be,  is  called 
a  trial  divisor,  since  the  next  term  of  the  root  is  obtained  from  the 
quotient.  The  term  of  the  root  sought  is  sometimes  less  than  the 
quotient,  since  the  dividend  may  contain  a  part  of  the  square  of  the 
next  term  of  the  root.  The  true  divisor  is  the  trial  divisor  with  the 
next  term  of  the  root  annexed. 

3.  If  the  number  is  not  a  perfect  square,  the  exact  root  can  not  be 
found.  The  exact  root  may  be  approximated  by  annexing  periods 
of  decimal  ciphers.  Since  the  square  of  no  one  of  the  nine  digits 
ends  with  a  cipher,  the  operation  may  be  continued  indefinitely. 

4.  In  pointing  off  a  decimal,  or  a  mixed  decimal  number,  begin 
with  the  order  of  units.  If  there  be  an  odd  number  of  decimal 
places,  annex  a  decimal  cipher. 

5.  When  both  terms  of  a  common  fraction  are  not  perfect  squares, 
the  exact  square  root  can  not  be  found.  An  approximate  root  may 
be  obtained  by  multiplying  both  terms  of  the  fraction  by  the  denom¬ 
inator,  and  extracting  the  root  of  the  resulting  fraction.  Thus, 


6.  The  square  root  of  a  perfect  square  may  be  found  by  resolving 
it  into  its  prime  factors,  and  taking  the  product  of  one  of  every  two 
of  those  that  are  equal. 


G-eometrical  Explanation. 


411.  The  area  of  a  square  surface  is  found  by  squaring 
the  length  of  one  side ;  and,  conversely,  the  length  of  the 
side  is  found  by  extracting  the  square  root  of  the  number 
denoting  the  area. 


256 


COMPLETE  ARITHMETIC, 


Let  the  annexed  diagram  represent  a  square  surface  whose 

area  is  625.  Required  the  length  of  one 
side. 

Since  the  number  denoting  the  area 
contains  two  periods,  there  are  two 
terms  in  the  square  root;  and  since  the 
greatest  square  in  the  left-hand  period 
is  4,  the  tens’  term  of  the  root  is  2. 
(Art.  409.)  Hence  the  length  of  the 
side  of  the  square  is  20  plus  the  units’  term  of  the  root. 
What  is  the  units’  term? 

Taking  from  the  given  surface  a  square  whose  side  is  20 

and  whose  area  is  400,  there  remains  a 
surface  whose  area  is  625  — 400,  or  225. 
This  surface  consists  of  two  equal  rect¬ 
angles,  each  20  in  length,  and  a  small 
square,  the  length  of  whose  side  equals 
the  width  of  each  rectangle.  What  is 
the  width  of  each  rectangle  ? 

Since  the  two  rectangles  contain  most 
of  the  surface  whose  area  is  225,  their  width  may  be  found 
by  dividing  225  by  their  joint  length,  which  is  twice  20,  or 
40.  The  quotient  is  5,  and  hence  the  width  of  each  rect¬ 
angle  is  5,  and  their  joint  area  is  40  X  5,  or  200. 

Removing  the  two  rectangles,  there  remains  the  small 

square,  whose  side  is  5  and  whose  area 
is  25,  the  difference  between  225  and 
200.  Hence,  5  is  the  units’  term  of 
the  root,  and  the  length  of  the  side  of 
the  square  is  20  -j-  5,  or  25. 

Adding  the  area  of  the  several  parts, 
we  have  202  -J-  20  X  5  X  2  -f-  52  =  400 
L—J  -f  200  -f  25  =  625. 

It  is  seen  that  the  square  whose  side  is  20,  represents  the 
square  of  the  tens  of  the  root;  the  two  rectangles,  twice  the 
product  of  the  tens  by  the  units;  and  the  smaller  square,  the 
square  of  the  unitSo 


CUBE  ROOT. 


257 


Note. — The  entire  length  of  the  surface  whose  area  is  225,  is  twice 
the  side  of  the  square  removed,  plus  the  side  of  the  smaller  square 
(20  X  2  -f-  5  =  45),  and  this  multiplied  by  5  gives  an  area  of  225. 


CUBE  ROOT. 


The  Division  of  the  Number  into  Periods. 

412.  The  cubes  of  the  smallest,  the  greatest,  and  an  inter¬ 
mediate  number,  composed  of  one,  two,  and  three  orders,  are 
as  follows  : 

l3  =  1  93  =  729  43  =  64 

103  =  1000  99 3  =  970299  443  =  85184 

1003  =  1000000  999  s  =  997002999  4443  =  87528384 

A  comparison  of  the  above  numbers  with  their  cubes 
shows  that  the  cube  of  a  number  contains  three  times  as 
many  orders  as  the  number,  or  three  times  as  many  orders 
less  two  or  less  one. 

413.  Hence,  if  a  number  be  separated  into  periods  of  three 
orders  each,  there  will  he  as  many  orders  in  its  cube  root  as 
there  are  'periods  in  the  number. 

1.  How  many  orders  in  the  cube  root  of  91125? 

Suggestion. — First  point  off  the  number  into  periods  of  three 
orders  each ;  thus,  9il25. 

2.  How  many  orders  in  the  cube  root  of  84604519? 

3.  How  many  orders  in  the  cube  root  of  912673? 

4.  How  many  orders  in  the  cube  root  of  48228544? 

5.  How  many  orders  in  the  cube  root  of  2357947691? 

414.  The  cubes  of  the  smallest  and  the  greatest  number 

of  units,  tens,  and  hundreds  are  as  follows : 

1»  =  1  93  =7  729 

103  =  1000  903  =  729000 

1003  =  1000000  9003  ==  729000000 

A  comparison  of  the  above  numbers  with  their  cubes 
shows  that  the  cube  of  units  gives  no  order  higher  than 

C.Ar.— 22. 


258 


COMPLETE  ARITHMETIC. 


hundreds ;  that  the  cube  of  tens  gives  no  order  lower  than 
thousands  nor  higher  than  hundred-thousands;  and  that 
the  cube  of  hundreds  gives  no  order  lower  than  millions 
nor  higher  than  hundred-millions. 

Hence,  if  a  number  be  separated  into  periods  of  three 
orders  each,  the  left-hand  period  will  contain  the  cube  of  the 
first  term  of  the  cube  root;  the  first  two  left-hand  periods  will 
contain  the  cube  of  the  first  two  terms  of  the  cube  root,  etc. 

6.  What  is  the  tens’  term  of  the  cube  root  of  91125? 

7.  What  is  the  tens’  term  of  the  cube  root  of  912673? 

8.  What  is  the  hundreds’  term  of  the  cube  root  of 
48228544? 

9.  What  is  the  first  term  of  the  cube  root  of  529475129? 

10.  What  is  the  first  term  of  the  cube  root  of  257259456? 

WRITTEN  PROBLEMS. 

11.  What  is  the  cube  root  of  262144? 

Since  262144  is  composed  of  two  pe¬ 
riods,  its  cube  root  will  be  composed  of 
two  orders  (Art.  413).  The  left-hand 
period,  262,  contains  the  cube  of  the  tens’ 
term  of  the  root  (Art.  414).  The  greatest 
cube  in  262  is  216,  the  cube  root  of  which 
is  6 ;  hence,  6  is  the  tens’  term  of  the  root.  How  is  the  units’  term 
to  be  found? 

The  cube  of  a  number,  composed  of  tens  and  units,  is  equal  to  the 
cube  of  the  tens,  plus  three  times  the  product  of  the  square  of  the 
tens  by  the  units,  plus  three  times  the  product  of  the  tens  by  the 
square  of  the  units,  plus  the  cube  of  the  units  (Art.  395).  Hence, 
the  difference  between  262144  and  the  cube  of  the  6  tens  of  its  cube 
root,  is  composed  of  three  times  the  product  of  the  square  of  the  tens  of  its 
root  by  the  units,  plus  three  times  the  product  of  the  tens  by  the  square  of  the 
units,  plus  the  cube  of  the  units. 

But  since  the  square  of  tens  gives  no  order  lower  than  hundreds 
(Art.  407),  the  461  hundreds  of  the  difference  (46144)  contains  three 
times  the  product  of  the  square  of  the  tens  by  the  units.  Hence,  if  the  461 
hundreds  (rejecting  the  two  right-hand  figures)  be  divided  by  three 


Process. 

262144 [64 
63  =  216  4 

62  X  3  =  108  )  461 44 
64 3  =  262144 


CUBE  ROOT. 


259 


times  the  square  of  the  6  tens  of  the  root,  the  quotient,  which  is  4, 
will  be  the  units’  teinn  of  the  root.  Cube  64,  and  subtract  the  result  from 
262144.  There  is  no  remainder,  and  hence  64  is  the  cube  root  sought. 

Note. — Instead  of  cubing  64,  the  parts  which  compose  the  differ¬ 
ence,  46144,  may  be  formed  and  added,  thus: 

602  X  4X3  =  43200 

60X4*  X  3=  2880 

43  — _ 64 

46144 

11.  What  is  the  cube  root  of  42875?  Of  91125? 

12.  What  is  the  cube  root  of  117649?  Of  185193? 

13.  What  is  the  cube  root  of  274625?  Of  405224? 

14.  What  is  the  cube  root  of  704969?  Of  912673? 

15.  What  is  the  cube  root  of  48228544? 

Process. 

48228544  [  364,  Cube  root. 

33  —  27  74,  Trial  quotients. 

32  X  3  —  27  )  212 
36 3  =  46656 
362  X  3  =  3888  )  15725 
364s  =48228544 

Since  the  two  right-hand  figures  of  each  dividend  are  rejected, 
only  the  first  figure  of  each  period  need  be  brought  down  and  an¬ 
nexed  to  the  difference. 

The  quotient  obtained  by  dividing  212  by  27  is  7,  which  is  too 
large  for  the  second  term  of  the  root,  since  the  cube  of  37  is  more 
than  48228,  the  first  two  periods. 

The  second  difference  is  found  by  subtracting  the  cube  of  36,  the 
first  two  terms  of  the  root,  from  48228,  the  first  two  periods  of  the 
number. 


16.  What  is  the  cube  root  of 

17.  What  is  the  cube  root  of 

18.  What  is  the  cube  root  of 

19.  What  is  the  cube  root  of 

20.  What  is  the  cube  root  of 

21.  What  is  the  cube  root  of 

22.  What  is  the  cube  root  of 


3048625  ?  Of  34328125  ? 
41063625?  Of  43614208? 
27270901  ?  Of  515849608  ? 
185193?  128024064? 
103823?  Of  27054036008? 
15.625?  Of  .074256? 
97.336?  Of  .015625? 


260 


COMPLETE  ARITHMETIC. 


23.  What  is  the  cube  root  of  56.47?  Of  12.3456? 

Suggestion. — Point  from  units’  order,  and  fill  decimal  periods, 
thus:  56.470,  and  12.345600. 

24.  What  is  the  cube  root  of  .000042875  ?  Of  67.917312  ? 

25.  What  is  the  cube  root  of  9  ?  Of  31  ?  Of  50  ? 

Suggestion. — Annex  periods  of  decimal  ciphers  and  carry  the 
root  to  three  decimal  places. 

26.  What  is  the  cube  root  of  2  ?  Of  20  ?  Of  200  ? 

27.  What  is  the  cube  root  of  yVA"?  Of  ? 

28.  What  is  the  cube  root  of  11-g-J?  Of 

29.  A  cubical  box  contains  19683  cubic  inches :  what  is 
the  length  of  its  edge  ? 

30.  A  block  of  granite  in  the  form  of  a  cube,  contains 
41063.625  cubic  inches  :  what  is  the  length  of  its  edge? 

31.  A  cubical  bin  holds  100  bushels :  what  is  the  length 
of  its  edge  ? 

32.  If  6  ranks  of  wood,  each  128  ft.  long,  3  ft.  wide,  and 
6  ft.  high,  were  piled  together  in  the  form  of  a  cube,  what 
would  be  the  height  of  the  pile  ? 

PRINCIPLES  AND  RULE. 

415.  Principles. — 1.  The  cube  root  of  a  number  contains 
as  many  orders  as  there  are  periods  of  three  figures  each  in  the 
number. 

2.  The  left-hand  period  of  a  number  contains  the  cube  of  the 
first  term  of  its  cube  root ;  the  two  left-hand  periods  contain  the 
cube  of  the  first  two  terms  of  the  cube  root,  etc. 

3.  The  cube  of  a  number,  composed  of  tens  and  units,  is 
equal  to  the  cube  of  the  tens,  plus  three  times  the  product  of  the 
square  of  the  tens  by  the  units,  plus  three  times  the  product  of 
the  tens  by  the  square  of  the  units,  plus  the  cube  of  the  units. 

416.  Rule. — To  extract  the  cube  root  of  a  number, 

1.  Begin  at  the  units ’  order  and  separate  the  number  into 
periods  of  three  orders  each. 

2.  Find  the  greatest  cube  in  the  left-hand  period,  and  place 
its  cube  root  at  the  right  for  the  first  term  of  the  root. 


CUBE  ROOT. 


261 


3.  Subtract  the  cube  of  the  first  term  of  the  root  from  the  left- 
hand  'period ,  and  to  the  difference  annex  the  first  figure  of  the 
next  period  for  a  dividend. 

4.  Take  three  times  the  square  of  the  first  term  oj  the  root  for 
a  trial  divisor,  and  the  quotient  for  the  second  term  of  the  root . 
Cube  the  root  now  found ,  and,  if  the  result  is  not  greater  than 
the  two  left-hand  periods ,  subtract,  and  to  the  difference  annex 
the  first  figure  of  the  next  period  for  a  second  dividend.  If  the 
cube  of  the  root  found  is  greater  than  the  two  left-hand  periods , 
diminish  the  second  term  of  the  root. 

5.  Take  three  times  the  square  of  the  two  terms  of  the  root 
found  for  a  second  trial  divisor,  and  the  quotient  for  the  third 
term  of  the  root.  Cube  the  three  terms  of  the  root  found,  and 
subtract  the  result  from  the  three  left-hand  periods,  and  continue 
the  operation  in  like  manner  until  all  the  terms  of  the  root  are 
found. 

Notes. — 1.  The  quotient  obtained  by  dividing  the  dividend  by  the 
trial  divisor  may  be  too  large,  since  three  times  the  square  of  the  next 
figure  of  the  root  may  be  a  part  of  the  dividend.  Usually  the  term 
of  the  root  sought  is  the  quotient,  or  one  less  than  the  quotient. 

2.  When  a  dividend  does  not  contain  the  trial  divisor,  write  a 
cipher  for  the  next  term  of  the  root.  Take  three  times  the  square  of 
the  root  thus  formed  for  a  trial  divisor,  and  to  the  dividend  annex 
the  two  remaining  figures  of  the  period,  and  the  first  figure  of  the 
next  period  for  a  new  dividend. 

3.  If  the  number  is  not  a  perfect  cube,  the  root  may  be  approx¬ 
imated  by  annexing  periods  of  decimal  ciphers,  thus  adding  decimal 
terms  to  the  root.  Sufficient  accuracy  is  usually  secured  by  continuing 
the  root  to  two  or  three  decimal  places. 

4.  When  both  terms  of  a  common  fraction  are  not  perfect  cubes, 
the  cube  root  may  be  found  approximately  by  multiplying  both  terms 
of  the  fraction  by  the  square  of  the  denominator,  and  extracting  the 
root  of  the  resulting  fraction.  The  error  will  be  less  than  one  di¬ 
vided  by  the  denominator  of  the  root. 

5.  The  above  methods  of  extracting  the  square  or  cube  root  of 
numbers,  is  a  general  method  by  which  any  root  may  be  extracted. 
The  fourth  root,  for  example,  is  found  by  dividing  the  number  into 
periods  of  four  figures  each,  then  taking  the  fourth  root  of  the  left- 
hand  period  for  the  first  term  of  the  root,  four  times  the  cube  of  this 
first  term  for  a  trial  divisor,  and  the  remainder  with  the  first  term  of 
the  next  period  annexed,  for  a  dividend,  etc. 

6.  The  cube  root  of  a  perfect  cube  may  be  found  by  resolving  it 
into  its  prime  factors  and  taking  the  product  of  one  of  every  three  of 
those  that  are  equal. 


262 


COMPLETE  ARITHMETIC. 


Oeome t rical  Explanation,  of1  fhe  Process  of  Ex 
tracting  the  Cube  Root. 


417.  Tlie  solid  contents  of  a  cube  are  found  by  cubing 

the  length  of  its  edge,  and,  con¬ 
versely,  the  length  of  the  edge  is 
found  by  extracting  the  cube  root 
of  the  number  denoting  the  solid 
contents. 

Let  the  annexed  cut  represent 
a  cube  whose  solid  contents  are 
15625.  Required  the  length  of 
the  edge. 


Process. 

I 5625 | 25 
2s  =  8 


202  X  3  =  1200  )  7625 
202  X  5  X  3  =  6000 
20  X  52  X  3  =  1500 
53  =  125 

7625 


6 


Since  the  number  denoting  the  solid  contents  contains 

two  periods,  there  will  be  two 
terms  in  the  cube  root,  and 
since  the  greatest  cube  in  the 
left-hand  period  is  8,  the  tens’ 
term  of  the  root  is  2  (Art.  414). 
Hence,  the  length  of  the  edge  of 
the  cube  is  20  plus  the  units’ 
term  of  the  root. 

What  is  the  units’  term  ? 


Taking  from  the  given  cube  a  cube  whose  edge  is  20  and 
whose  capacity  is  8000,  there  remains  a  solid  whose  capacity 


is  15625  —  8000,  which  is  7625.  An  inspection  of  the  an¬ 
nexed  cut  shows  that  this  solid  contains  three  equal  rectan-  * 


CUBE  ROOT, 


263 


gular  solids,  whose  inner  face  (202)  is  equal  to  the  face  of 
the  removed  cube  and  whose  thickness  equals  the  units’ 
term  of  the  root.  What  is  the  thickness  of  each  of  these 
rectangular  solids? 

Since  they  compose  only  a  part  of  the  solid  whose  solid 
contents  are  7625,  their  thickness  can  not  be  greater  than 
the  quotient  obtained  by  dividing  7625  by  the  area  of  their 
joint  inner  faces,  which  is  202  X  3,  or  1200.  The  quotient 
is  6,  which  is  at  least  one  greater  than  the  thickness  of  each 
of  the  three  rectangular  solids,  since  263  is  greater  than 
15625,  the  solid  contents  of  the  given  cube.  Try  5  for  the 
thickness.  253  =  15625,  and  hence  5  is  the  required  thick¬ 
ness,  and  the  length  of  the  edge  of  the  given  cube  is  20  -f-  5, 
or  25. 

The  correctness  of  this  result  may  also  be  shown  by  find¬ 
ing  the  solid  contents  of  the  several  parts  of  the  given  cube. 
The  solidity  of  the  cube  removed  is,  as  shown  above,  203  = 
8000.  The  joint  solidity  of  the  three  adjacent  rectangular 
solids  is  202  X  5  X  3  =  6000. 

Removing  these  three  rectan¬ 
gular  solids,  there  remain  three 
other  rectangular  solids,  whose 
solidity  is  20  X  52,  or  500  each, 
and  whose  combined  solidity  is 
500  X  3,  or  1500. 

Removing  these  three  rectangular  solids,  there  remains 
the  small  cube,  whose  solidity  is 
53  ==  125. 

Adding  the  solidity  of  the  sev¬ 
eral  parts,  we  have  8000  -\~  6000 
+  1500  +  125  =  15625,  which  is 
the  solidity  of  the  given  cube. 

It  is  seen  that  the  cube  whose 
edge  is  20,  represents  the  cube  of 
.  the  tens  of  the  root;  the  three  ad- 


264 


COMPLETE  ARITHMETIC. 


jacent  rectangular  solids  represent  three  times  the  product  of 
the  square  of  tens  hy  the  units;  the  three  smaller  rectangular 
solids,  three  times  the  product  of  the  tens  by  the  square  of  the 
units;  and  the  smaller  cube,  the  cube  of  the  units. 


MENSURATION,  INVOLVING  INVOLUTION  AND 

EVOLUTION. 


I.  THE  RIGHT-ANGLED  TRIANGLE. 


418.  The  Hypotenuse  of  a  right-angled  triangle  is 
the  side  opposite  the  right  angle.  The  other  two  sides  are 
called  the  Base  and  the  Perpendicular.  (Art.  155.) 


419.  Principles. 


— 1.  The  square  of  the  hypotenuse  of  a 
right-angled  triangle  is  equal  to  the  sum 
of  the  squares  of  the  other  two  sides. 

This  principle,  which  may  be  proven  by 
geometry,  is  illustrated  by  the  annexed 
diagram. 

2.  The  square  of  the  base  or  the  per¬ 
pendicular  of  a  right-angled  triangle  is 
equal  to  the  square  of  the  hypotenuse  less 
the  square  of  the  other  side. 


PROBLEMS. 

1.  The  base  of  a  right-angled  triangle  is  8,  and  the  per¬ 
pendicular  6 :  what  is  the  length  of  the  hypotenuse  ? 

Solution. — Since  the  square  of  the  hypotenuse  equals  the  square 
of  the  base  plus  the  square  of  the  perpendicular,  the  hypotenuse 
equals  V 82  +  62  =  V 100  =  10. 

2.  The  hypotenuse  of  a  right-angled  triangle  is  20  inches 
and  the  base  is  16  inches:  what  is  the  perpendicular? 

3.  The  hypotenuse  of  a  right-angled  triangle  is  45  feet 
and  the  perpendicular  is  27  feet:  what  is  the  base? 


APPLICATIONS  OF  INVOLUTION  AND  EVOLUTION.  265 

4.  A  rectangular  field  is  192  yards  long  and  144  yards 
wide:  what  is  the  length  of  the  diagonal? 

5.  The  foot  of  a  ladder  is  18  feet  from  the  base  of  a 
building,  and  the  top  reaches  a  window  24  feet  from  the 
base :  what  is  the  length  of  the  ladder  ? 

6.  Two  boys  start  from  the  same  point,  and  one  walks  96 
rods  due  north,  and  the  other  72  rods  due  east :  how  far  are 
they  apart? 

7.  A  flag  pole  180  feet  high  casts  a  shadow  135  feet  in 
length :  what  is  the  distance  from  the  top  of  the  pole  to  the 
end  of  the  shadow? 

8.  A  boy  in  flying  his  kite  let  out  240  feet  of  string,  and 
the  distance  from  where  he  stood  to  a  point  directly  under 
the  kite  was  208  feet :  how  high  was  the  kite,  supposing  the 
string  to  be  straight? 

9.  A  rectangular  field  is  84  rods  long  and  63  rods  wide : 
what  is  the  side  of  a  square  field  of  the  same  area? 

10.  A  farm  is  125  rods  square,  and  a  rectangular  farm, 
containing  the  same  number  of  acres,  is  150  rods  in  length : 
what  is  its  width? 

420.  Rules. — 1.  To  find  the  hypotenuse  of  a  right-angled 
triangle,  Extract  the  square  root  of  the  sum  of  the  squares  of 
the  other  two  sides. 

2.  To  find  the  base  or  the  perpendicular  of  a  right-angled 
triangle,  Extract  the  square  root  of  the  difference  between 
the  squares  of  the  hypotenuse  and  the  other  side. 

II.  THE  CIRCLE. 

421.  Principles. — 1.  The  area  of  a  circle  is  equal  to  the 
square  of  its  diameter  multiplied  by  .7854.  Hence, 

2.  The  diameter  of  a  circle  equals  the  square  root  of  the 
quotient  of  the  area  divided  by  .7854. 

3.  The  areas  of  two  circles  are  to  each  other  as  the  squares 
of  their  diameters. 

Note. — The  above  propositions  can  be  proved  by  geometry. 

C.  Ar.—  23. 


266 


COMPLETE  ARITHMETIC. 


PROBLEMS. 

11.  The  diameter  of  a  circle  is  15  inches:  what  is  its 
area? 

12.  A  circular  pond  is  100  feet  in  diameter:  how  many 
square  yards  does  it  contain  ? 

13.  A  circular  room  has  an  area  of  78.54  square  yards: 
what  is  its  diameter? 

14.  How  many  circles,  each  3  inches  in  diameter,  will 
equal  in  area  a  circle  whose  diameter  is  2  feet? 

15.  How  many  circles,  each  15  inches  in  .diameter,  will 
equal  in  area  a  circle  whose  diameter  is  5  feet? 

16.  A  horse,  tied  to  a  stake  by  a  rope,  can  graze  to  the 
distance  of  40  feet  from  the  stake :  on  how  much  surface 
can  it  graze? 

17.  A  horse,  tied  to  a  stake,  can  graze  on  218|-  square 
yards  of  surface :  to  what  distance  from  the  stake  can  it 
graze  ? 

18.  How  many  circles,  each  3  inches  in  diameter,  contain 
the  same  area  as  a  surface  2.5  feet  square? 

III.  THE  SPHERE. 

422.  Principles. — 1.  The  surface  of  a  sphere  is  equal  to 
the  square  of  the  diameter  multiplied  by  3.1416. 

2.  The  solidity  of  a  sphere  is  equal  to  the  cube  of  the  diameter 
multiplied  by  .5236. 

3.  Two  spheres  are  to  each  other  as  the  cubes  of  their  diame¬ 
ters. 

Note. — The  surface  of  a  sphere  may  also  be  found  by  multiplying 
the  circumference  by  the  diameter;  and  the  solidity  by  multiplying  the 
surface  by  one  sixth  of  the  diameter. 

PROBLEMS. 

19.  What  is  the  surface  of  a  sphere  whose  diameter  is  10 
inches  ? 

20.  How  many  square  miles  on  the  surface  of  the  earth, 
its  mean  diameter  being  7912  miles? 


APPLICATIONS  OF  INVOLUTION  AND  EVOLUTION.  267 


21.  How  many  cubic  miles  in  the  solidity  of  the  earth? 

22.  How  many  cubic  inches  in  a  cannon  ball  whose 
diameter  is  7  inches? 

23.  How  many  balls  2  inches  in  diameter,  equal  in  solid¬ 
ity  a  ball  whose  diameter  is  8  inches? 

24.  The  diameter  of  the  earth  is  about  4  times  the  diam¬ 
eter  of  the  moon :  how  many  times  larger  than  the  moon  is 
the  earth? 

25.  The  diameter  of  Jupiter,  the  largest  planet,  is  about 
85000  miles,  and  the  diameter  of  the  sun  is  about  850000 
miles:  how  many  times  larger  than  Jupiter  is  the  sun? 

26.  The  surface  of  the  planet  Mercury  contains  about 
28274400  square  miles:  what  is  its  diameter? 

27.  The  planet  Uranus  contains  about  18816613200000 
cubic  miles :  what  is  its  diameter  ? 

Suggestion. — Divide  the  solidity  by  .5236,  and  extract  the  cube 
root  of  the  quotient. 

28.  A  brass  ball  contains  904.7808  cubic  inches:  what  is 
its  diameter? 


29.  A  square  and  a  triangle  contain  an  equivalent  area, 
and  the  base  of  the  triangle  is  36.1  inches,  and  its  altitude 
is  5  inches :  what  is  the  side  of  the  square  ? 

30.  One  of  the  mammoth  pines  of  California  is  110  feet 
in  circumference :  what  is  its  diameter  ? 

31.  How  many  cubic  feet  in  a  portion  of  the  above  tree 
100  feet  in  length,  supposing  its  mean  circumference  to  be 
94^  feet? 

32.  The  mean  distance  of  the  earth  from  the  sun  (new 
value)  is  about  91400000  miles,  and  it  revolves  in  its  orbit 
in  365^  days :  what  is  its  mean  hourly  motion  ? 

33.  The  mean  distance  of  Mercury  from  the  sun  (new 
value)  is  about  35400000  miles,  and  it  revolves  in  its  orbit 
in  87.9  days:  what  is  its  mean  hourly  motion? 

34.  The  diameter  of  the  moon  is  about  2000  miles :  how 
does  the  extent  of  the  moon’s  surface  compare  with  that  of 
the  earth,  whose  diameter  is  about  8000  miles? 


268 


COMPLETE  ARITHMETIC. 


SECTION  XVII. 

GENERAL  REVIEW. 

Note. — The  following  problems  are  selected  from  several  sets  used 
in  the  examination  of  pupils  for  promotion  to  high  schools,  and  in  the 
examination  of  teachers.  Problems  marked  *  are  simplified  in  this 
edition. 

MENTAL  PROBLEMS. 

1.  If  3  apples  are  worth  2  oranges,  how  many  oranges 
are  24  apples  worth? 

2.  How  long  will  it  take  a  man  to  lay  up  $60,  if  he  earn 
$15  a  week  and  spend  $9? 

3.  -J  of  74J  is  f  of  what  number? 

4.  |  of  45  is  -§  of  how  many  times  10? 

5.  A  has  20  cents ;  and  f  of  what  A  has  is  f  of  what 
B  has:  how  many  has  B? 

6.  If  -J  of  a  yard  of  cloth  cost  63  cents,  what  will  f  of 
a  yard  cost? 

7.  If  3  yards  of  muslin  cost  13J  cents,  what  will  j-  of  a 
yard  cost? 

8.  The  difference  between  f  and  -g-  of  a  number  is  10: 
what  is  the  number? 

9.  What  fraction  is  as  much  greater  than  §  as  -J  is  less? 

10.  A  piece  of  flannel  lost  J-  of  its  length  by  shrinkage 
in  fulling,  and  then  measured  30  yards :  what  was  its  length 
before  fulling? 

11.  A  horse  cost  $90,  and  t3q-  of  the  price  of  the  horse 
equals  f  of  3  times  the  cost  of  the  saddle:  what  did  the 
saddle  cost? 

12.  If  to  my  age  you  add  its  half,  its  third,  and  28  years, 
the  sum  will  be  three  times  my  age :  what  is  my  age  ? 

13.  A  boy  being  asked  his  age,  said  that  f  of  80  was  -f 
of  10  times  his  age:  what  was  his  age? 


GENERAL  REVIEW. 


269 


14.  A  boy  gave  f  of  his  money  for  a  sled,  -J  of  it  for  a 
hat,  and  then  had  7  cents  left:  how  many  cents  had  he  at 
first  ? 

15.  |  of  my  money  is  in  my  purse,  |  in  my  hand,  and 
the  remainder,  which  is  25  cents,  is  in  my  pocket:  how 
much  money  have  I  ? 

16.  A  boy  having  f  of  a  dollar,  gave  of  his  money  to 

John  and  ^  of  the  remainder  to  James:  what  part  of  a 

dollar  did  James  receive? 

17.  A  farmer  sold  -§  of  his  sheep  and  then  bought  f  as 

many  as  he  had  left,  when  he  had  40  sheep :  how  many  had 

he  at  first? 

18.  John  lost  f  of  his  money  and  spent  -J-  of  the  re¬ 
mainder,  and  then  had  only  10  cents :  how  much  money 
had  he  at  first? 

19.  A  man  sold  a  horse  for  $60,  which  was  %  of  f  of  its 
cost :  how  much  was  lost  by  the  bargain  ? 

20.  A  man  sold  a  horse  for  $130,  which  was  f  more  than 
it  cost  him :  what  was  the  cost  of  the  horse  ? 

21.  A  sold  B  a  horse  for  ^  more  than  its  cost,  and  B  sold 
it  for  $80,  losing  ^  of  its  cost :  how  much  did  A  pay  for  the 
horse  ? 

22.  At  $^  a  bushel,  how  many  bushels  of  corn  may  be 
bought  for  $8? 

23.  If  J  of  a  bushel  of  wheat  cost  $f,  what  part  of  a 
bushel  can  be  bought  for  $f  ? 

24.  If  $18f  will  purchase  f  of  a  load  of  corn,  what  part 
of  it  will  $16f  purchase? 

25.  If  2-|-  pounds  of  cheese  cost  3J  dimes,  what  part  of  a 
pound  can  be  bought  for  1  dime? 

26.  How  many  bushels  of  coal  at  12-%  cents  a  bushel  can 
be  bought  for  $15? 

27.  What  part  of  7  bushels  is  -f  of  a  peck  ? 

28.  What  part  of  a  pound  of  gold  is  .25  of  an  ounce? 

29.  What  part  of  f  of  a  gallon  is  f  of  a  pint? 

30.  From  f  of  a  day  take  -J  of  an  hour. 

31.  If  a  staff  5  feet  long  cast  a  shadow  2  feet  long  at  12 


270 


COMPLETE  ARITHMETIC. 


o’clock,  what  is  the  height  of  a  steeple  whose  shadow,  at  the 
same  hour,  is  80  feet? 

32.  If  a  five  cent  loaf  weigh  10  ounces  when  flour  is  $4 
a  barrel,  what  ought  it  to  weigh  when  flour  is  $5  a  barrel  ? 

33.  If  20  bushels  of  oats  will  feed  40  horses  80  days,  how 
long  will  180  bushels  feed  them? 

34.  If  a  horse  eat  2  bushels  of  oats  in  6  days,  in  how 
many  days  will  2  horses  eat  18  bushels? 

35.  If  3  men  can  mow  18  acres  of  grass  in  4  days,  how 
many  men  can  mow  9  acres  in  3  days? 

36.  A  garrison  of  20  men  is  supplied  with  provisions  for 
12  days :  if  12  men  leave,  how  long  will  the  provisions  serve 
the  remainder? 

37.  A  man  bought  a  watch  and  chain  for  $80,  and  the  chain 
cost  ^  as  much  as  the  watch:  how  much  did  each  cost? 

38.  A  has  1J  times  as  many  cents  as  B,  and  they  together 
have  40  cents :  how  many  has  each  ? 

39.  A  pole  120  feet  high  fell  and  broke  into  two  parts, 
and  J-  of  the  longer  part  was  equal  to  the  shorter :  how  long 
was  each  part? 

40.  A  and  B  together  own  824  sheep,  and  A  has  If  times 
as  many  as  B :  how  many  has  each  ? 

41.  A,  B,  and  C  rent  a  pasture  for  $42 ;  B  pays  half  as 
much  as  A,  and  C  half  as  much  as  B :  what  does  each 
pay? 

42.  A  and  B  own  a  farm ;  A  owns  f  as  much  as  B,  and 
B  owns  40  acres  more  than  A:  how  many  acres  does  each 
own? 

43.  f  of  A’s  money  is  §  of  B’s,  and  f  of  B’s  is  j  of  C’s, 
which  is  $81 :  how  much  have  A  and  B  each  ? 

44.  If  a  man  can  reap  f  of  an  acre  of  wheat  in  a  day, 
how  much  can  6  men  reap  in  10  days? 

45.  A  makes  a  shoe  in  f  of  a  day ;  B  makes  one  in  f  of 
a  day:  how  many  shoes  can  both  make  in  a  day? 

46.  A  can  mow  an  acre  of  grass  in  f  of  a  day,  and  B  in 
f  of  a  day :  how  long  will  it  take  both  together  to  mow  an 
acre? 


GENERAL  REVIEW. 


271 


47.  A  can  mow  a  field  of  grass  in  5  days,  and  B  in  4  days : 
how  long  will  it  take  both,  working  together,  to  mow  it? 

48.  A  can  build  a  house  in  20  days,  but,  with  the  assist-  • 
ance  of  C,  he  can  do  it  in  12  days :  in  what  time  can  C  do 
it  alone  ? 

49.  A  alone  can  build  a  certain  wall  in  6  days,  B  alone 
in  10  days,  and  C  alone  in  15  days :  in  how  many  days  can 
they  all  together  build  it? 

50.  A,  B,  and  C  can  do  a  job  in  20  days ;  A  and  B  can 
do  it  in  40  days ;  and  A  and  C  in  30  days :  in  how  many 
days  can  each  do  it  alone? 

51.  A  broker  bought  rail-road  stock  at  80  and  sold  it  at 
70:  what  per  cent  did  he  lose? 

52.  A  broker  bought  stock  at  70  and  sold  it  at  90:  what 
per  cent  did  he  gain  ? 

53.  A  merchant  bought  40  yards  of  cloth  for  $90 :  at  how 
much  a  yard  must  he  sell  it  to  gain  33^  per  cent? 

54.  For  how  much  must  tea  costing  90  cents,  be  sold  to 
gain  12^  per  cent  ? 

55.  A  man  bought  a  hat  for  $5  and  sold  it  for  $6  :  what 
per  cent  did  he  gain? 

56.  I  sell  cloth  at  $2.50  a  yard  and  gain  25  per  cent: 
for  how  much  must  I  sell  it  to  lose  20  per  cent  ? 

57.  A  man  earned  a  certain  sum  of  money,  and,  after 
adding  to  it  $12.50,  found  that  what  he  then  had  was  133^ 
per  cent  of  what  he  earned :  how  much  did  he  earn  ? 

58.  A  man  sold  a  watch  for  $90,  and  gained  50  per  cent : 
wThat  per  cent  wrould  he  have  gained  if  he  had  sold  it  for 
$75? 

59.  f  of  the  price  received  for  an  article  is  equal  to  |  of 
its  cost:  what  is  the  gain  per  cent? 

60.  Two  men,  A  and  B,  engaged  in  trade  with  different 
capitals ;  A  lost  33^  per  cent  of  his  capital,  and  B  gained 
50  per  cent  on  his,  when  each  had  $600:  with  what  capital 
did  each  begin  trade  ? 

61.  How  much  grain  must  I  take  to  mill  to  bring  away 
2  bushels  after  the  miller  lias  taken  10  per  cent  for  toll? 


272 


COMPLETE  ARITHMETIC. 


62.  At  what  rate  per  cent,  simple  interest,  will  $1  double 
itself  in  8  years? 

63.  The  interest  on  a  certain  sum  for  4  years  was  ^  the 
sum :  what  was  the  rate  per  cent  ? 

64.  Two  men  start  from  two  places  495  miles  apart,  and 
travel  toward  each  other;  one  travels  20  miles  a  day,  and 
the  other  25  miles  a  day:  in  how  many  days  will  they 
meet  ? 

65.  A  owes  -f  of  B’s  income,  but,  by  saving  of  B’s 
income  annually,  he  can  pay  his  debt  in  5  years,  and  have 
$50  left:  what  is  B’s  income? 

66.  C  and  D  are  traveling  in  the  same  direction  ;  C  is 
18  miles  ahead  of  D,  but  D  travels  7  miles  while  C  travels 
4 :  how  many  miles  from  the  place  of  starting  will  D  have 
traveled  when  he  overtakes  C? 

67.  If  a  man  traveling  14  hours  a  day,  perform  half  a 
journey  in  5  days,  how  long  will  it  take  to  perform  the 
other  half,  if  he  travel  10  hours  a  day? 

68.  If  a  man  can  do  a  piece  of  work  in  days,  working 
8  hours  a  day,  how  long  will  it  take,  if  he  work  6  hours  a 
day? 

*69.  A  is  20  years  of  age;  4  times  A’s  age  equals  the 
sum  of  B’s  and  C’s  ages ;  and  C’s  age  is  f  of  B’s  age :  what 
are  the  ages  of  B  and  C? 

70.  A  hare  is  30  rods  before  a  hound,  but  the  hound 
runs  7  rods  while  the  hare  runs  5 :  how  far  must  the  hound 
run  to  catch  the  hare? 

*71.  A  hare  starts  50  leaps  before  a  hound,  and  leaps  4 
times  while  the  hound  leaps  3  times;  but  1  of  the  hound’s 
leaps  equal  2  of  the  hare’s :  how  many  leaps  must  the  hound 
take  to  gain  one  of  the  hare’s  leaps?  To  catch  the  hare? 

*72.  If  a  steamer  sails  9  miles  an  hour  down  stream,  and 
5  miles  an  hour  up  stream,  how  many  hours  will  it  be  in 
sailing  45  miles  down  stream  and  returning? 

73.  A  steamer  sails  a  mile  down  stream  in  5  minutes, 
and  a  mile  up  stream  in  7  minutes :  how  far  down  stream 
can  it  go  and  return  again  in  one  hour? 


GENERAL  REVIEW 


‘273 

74.  A  pipe  will  fill  a  cistern  in  4  hours,  and  another 
will  empty  it  in  6  hours :  how  long  will  it  take  to  fill  it 
when  both  pipes  run  ? 

75.  At  what  time  between  six  and  seven  o’clock  are  the 
hour  and  minute  hands  of  a  clock  together  ? 

WRITTEN  PROBLEMS. 

76.  The  minuend  is  1250,  and  the  remainder  592 :  what 
is  the  subtrahend  ? 

77.  The  quotient  is  71,  the  divisor  42,  and  the  remainder 
15  :  what  is  the  dividend  ? 

78.  If  a  certain  number  be  multiplied  by  22,  and  64  be 
subtracted  from  the  product,  and  the  remainder  be  divided 
by  4,  the  quotient  will  be  50:  what  is  the  number? 

79.  What  will  be  the  cost  of  3760  lbs.  of  hay,  at  $8.50  a 
ton  ? 

80.  At  $24.50  per  acre,  how  many  acres  of  land  can  be 
bought  for  $3560.75? 

81.  Add  -J,  -J,  •§•  of  f-,  and  -f  of 

82.  From  17^-  take  |  of  6^,  and  multiply  the  remainder 
by  -§. 

83.  Multiply  f  of  -§  by  -J-  of  f,  and  divide  the  product 

by  A- 

84.  Divide  f  of  6^  by  -§  of  7^. 

85.  What  number  multiplied  by  28f  will  produce  145? 

86.  From  the  sum  of  215§  and  125J  take  their  differ¬ 
ence. 

87.  Multiply  f  -f  f  of  J  by  f  —  -f  of  f . 

88.  Divide  2|-  by  3J,  and  multiply  the  quotient  by  3^. 

89.  What  must  8f|  be  multiplied  by  that  the  product 
may  be  3  ? 

90.  A  man  bought  of  a  section  of  land  for  $2880,  and 
sold  -J  of  it  at  $10  an  acre,  and  the  rest  at  $12  an  acre: 
how  much  did  he  gain? 

91.  A  merchant  owning  -J  of  a  ship  sells  -f  of  his  share 
for  $16800 :  what  is  the  value  of  the  whole  ship,  at  this 
rate,  and  what  part  of  the  ship  has  he  left? 


274 


COMPLETE  ARITHMETIC. 


92.  Add  9  thousandths,  3  hundredths,  and  7  units. 

93.  From  15  ten-thousandths  take  27  millionths,  and 
multiply  the  difference  by  20.5. 

94.  Multiply  160  by  .016,  and  divide  the  product  by 
.0025. 

95.  Multiply  15  thousandths  by  15  hundredths,  and  from 
the  product  take  15  millionths. 

96.  Divide  256  thousandths  by  16  millionths. 

97.  Multiply  625  by  .003,  and  divide  the  result  by  25. 

98.  Change  yfo  to  a  decimal,  and  divide  the  result  by  2J. 

99.  Change  yfy  to  a  decimal,  and  divide  the  result  by 
5000. 

100.  Reduce  .625  of  a  pound  Troy  to  lower  integers. 

101.  What  decimal  of  a  rod  is  .165  of  a  foot? 

102.  What  will  63  thousandths  of  a  cord  of  wood  cost, 
at  $2.25  per  cord  ? 

103.  How  many  minutes  were  there  in  the  month  of 
February,  1880? 

104.  How  many  seconds  are  there  in  the  three  summer 
months  ? 

105.  How  many  steps,  2  ft.  4  in.  each,  will  a  person 
take  in  walking  miles? 

106.  How  many  times  will  a  wheel,  12  ft.  6  in.  in  cir¬ 
cumference,  turn  round  in  rolling  one  mile? 

107.  How  many  acres  in  a  street  4  rods  wide  and  2^ 
miles  long? 

108.  How  many  yards  of  carpeting,  f  of  a  yard  wide, 
will  it  take  to  cover  a  parlor,  18J  feet  long  and  15  feet 
wide  ? 

109.  How  many  grains  in  14  ingots  of  silver,  each  weigh¬ 
ing  27  oz.  10  pwt.  ? 

110.  How  many  square  feet  of  lumber  in  40  boards,  each 
12  feet  long  and  7^-  inches  wide? 

111.  What  will  a  board  20  feet  long  and  9  inches  wide 
cost,  at  $30  a  thousand? 

112.  What  will  it  cost  to  lay  a  pavement  36  feet  long 
and  9  feet  6  inches  wide,  at  40  cents  a  square  yard? 


GENERAL  REVIEW. 


275 


113.  A  pile  of  wood,  containing  10  cords,  is  20  feet  long 
and  8  feet  wide :  how  high  is  it  ? 

114.  What  is  the  value  of  a  pile  of  wood  40  feet  long,  8 
feet  wide,  and  5J  feet  high,  at  $5.30  a  cord? 

115.  How  many  sacks,  holding  2  bu.  3  pk.  2  qt.  each, 
can  be  filled  from  a  bin  containing  366  bu.  3  pk.  4  qt.  of 
wheat  ? 

116.  A  lady  bought  6  silver  spoons,  each  weighing  3  oz. 
3  pwt.  8  gr.,  at  $2.25  an  ounce,  and  a  gold  chain,  weighing 
14  pwt.,  at  $1.25  a  pwt. :  what  was  the  cost  of  both  spoons 
and  chain? 

117.  How  many  bricks  will  it  require  to  build  a  wall  2 
rods  long,  6  feet  high,  and  18  inches  thick,  each  brick  being 
8  inches  long,  4  inches  wide,  and  2 \  inches  thick? 

118.  Cincinnati  is  7°  49'  west  of  Baltimore:  when  it  is 
noon  at  Baltimore,  what  is  the  hour  at  Cincinnati? 

119.  New  York  is  75  degrees  of  longitude  west  of  Lon¬ 
don  :  when  it  is  noon  at  New  York  what  is  the  hour  at 
London  ? 

120.  Boston  is  71°  4'  9"  W.  longitude,  and  Cleveland  is 
81°  47'  W. :  when  it  is  4  P.  M.  at  Cleveland,  what  is  the 
hour  at  Boston  ? 

121.  What  part  of  a  rod  is  2  ft.  9  in.  ? 

122.  Keduce  5  fur.  8  rd.  to  the  decimal  of  a  mile. 

123.  Reduce  f  of  a  square  yard  to  the  fraction  of  an 
acre. 

124.  From  J-  of  a  pound  Troy  take  f  of  an  ounce. 

125.  Reduce  f  of  a  quart  to  the  fraction  of  a  bushel. 

126.  A  regiment  lost  8  %  of  its  men  in  a  battle,  and 
25%  of  those  that  remained  died  from  sickness,  and  it  then 
mustered  621  men:  how  many  men  were  in  the  regiment  at 
first  ? 

127.  A  quantity  of  sugar  was  bought  for  $150,  and  sold 
for  $167.50:  what  was  the  gain  per  cent? 

128.  A  merchant  bought  500  yards  of  cloth  for  $1800 : 
for  how  much  a  yard  must  he  sell  it  to  gain  25  %  ? 

129.  A  man  sold  a  piece  of  cloth  for  $24,  and  thereby 


276 


COMPLETE  ARITHMETIC. 


lost  25  % ;  if  he  had  sold  it  for  $34,  would  he  have  gained 
or  lost,  and  what  per  cent  ? 

130.  I  sold  goods  at  20  %  gain,  and,  investing  the  pro¬ 
ceeds,  sold  at  20  %  loss :  did  I  gain  or  lose  by  the  opera¬ 
tion,  and  what  per  cent  ? 

131.  Sold  2  carriages,  at  $240  apiece,  and  gained  20  % 
on  one  and  lost  20  %  on  the  other :  how  much  did  I  gain 
or  lose  in  the  transaction? 

132.  A  man  bought  a  horse  for  $72,  and  sold  it  for  25  % 
more  than  cost,  and  10  %  less  than  he  asked  for  it :  what 
did  he  ask  for  it? 

133.  A  merchant  marked  a  lot  of  goods,  costing  $5800, 
at  30  %  above  cost,  but  sold  them  at  10  %  less  than  the 
marked  price :  how  much  and  what  per  cent  did  he 
gain  ? 

134.  What  must  I  ask  for  cloth,  costing  $4  a  yard,  that 
I  may  deduct  20  %  from  my  asking  price  and  still  make 
20  %? 

135.  A  man  bought  stock  at  25%  below  par  and  sold  it 
at  20  %  above  par :  what  per  cent  did  he  make  ? 

136.  A  fruit  dealer  lost  33J  per  cent  of  a  lot  of  apples, 
and  sold  the  remainder  at  a  gain  of  50  per  cent :  required 
the  per  cent  of  gain  or  loss. 

137.  I  bought  63  kegs  of  nails,  each  keg  containing 
100  lbs.,  at  4r^  cents  a  pound,  and  sold  -§  of  them  for  what 

of  them  cost:  what  per  cent  did  I  lose  on  the  part  sold? 

138.  I  bought  $128.25  worth  of  goods;  kept  them  on 
hand  6  months  when  money  was  worth  8  %  interest,  and 
then  sold  them  at  a  net  gain  of  6  % :  for  how  much  were 
they  sold? 

139.  When  money  was  worth  9%  interest,  I  bought  $800 
worth  of  goods,  kept  them  4  months  and  then  sold  them 
for  $959.10:  what  per  cent  on  the  cost  did  I  gain? 

140.  A  house  valued  at  $3240  is  insured  for  -J  of  its 
value,  at  f  %:  what  is  the  premium? 

141.  I  pay  $19.20  premium  for  insuring  my  house  for  f 
of  its  value,  at  1^-  %  :  what  is  the  value  of  my  house  ? 


GENERAL  REVIEW. 


277 


142.  A  capitalist  sent  a  broker  $25000  to  invest  in  cotton, 
after  deducting  his  commission  of  2^-%:  how  much  cotton, 
at  5  cents  a  pound,  did  the  broker  purchase  ? 

143.  An  agent  received  $502.50  to  purchase  cloth,  after 
deducting  \  %  commission  :  how  many  yards  did  he  buy 
at  $1.25  a  yard? 

144.  What  is  the  interest  of  $125.50  for  7  months  and 
10  days,  at  7  %  ? 

145.  What  is  the  interest  of  $50000  for  one  day,  at  8%? 

146.  What  is  the  interest  of  $75.50  from  June  12,  1869, 
to  Aug.  6,  1870,  at  7^  %  ? 

147.  A  man  loaned  $800  for  2  years  and  6  months,  and 
received  $90  interest :  what  was  the  rate  per  cent  ? 

148.  At  what  rate  per  cent  will  $311.50  amount  to 
$337.40  in  1  yr.  4  mo.  ? 

149.  What  sum  of  money  will  yield  as  much  interest  in 
3  years,  at  per  cent,  as  $540  yields  in  1  yr.  8  mo.,  at 
7%? 

150.  The  amount  of  a  certain  principal  for  3  years,  at  a 
certain  rate  per  cent,  is  $750,  and  the  interest  is  J  of  the  prin¬ 
cipal  :  what  is  the  principal,  and  what  is  the  rate  percent? 

151.  A  note  for  $500,  dated  Oct.  8,  1864,  and  bearing 
interest  at  9  %,  is  indorsed  as  follows:  Nov.  4,  1865,  $30; 
Jan.  30,  1866,  $250.  What  was  due  July  1,  1866? 

152.  What  is  the  present  worth  of  a  note  of  $1320,  due 
in  3  years  and  4  months,  without  interest,  money  being 
worth  6  %  ?  What  is  the  discount  ? 

153.  What  is  the  true  discount  of  $236,  due  in  3  years, 
at  6  %  ? 

154.  What  is  the  bank  discount  on  $125,  payable  in  90 
days,  at  8  %  ? 

155.  What  is  the  difference  between  the  true  discount 
and  the  bank  discount  of  $359.50,  for  90  days,  without 
grace,  at  12  %  ? 

156.  For  what  sum  must  I  give  my  note  at  a  bank,  pay¬ 
able  in  4  months,  at  10  %,  to  get  $300? 

157.  I  borrow  of  A  $150  for  6  months,  and  afterward  I 


278 


COMPLETE  ARITHMETIC. 


lend  him  $100:  how  long  may  he  keep  it  to  balance  the 
use  of  the  sum  he  lent  me? 

158.  A  owes  B  $300,  of  which  $50  is  due  in  2  months, 
$100  in  5  months,  and  the  remainder  in  8  months :  what  is 
the  equated  time  for  the  whole  sum? 

159.  A  man  owes  $300  due  in  5  months,  and  $700  due  in 
3  months,  and  $200  due  in  8  months :  if  he  pays  \  of  the 
whole  in  2  months,  when  ought  the  other  half  to  be  paid? 

160.  I  have  sold  50  bushels  of  wheat  for  A,  and  60 
bushels  for  B,  receiving  $150  for  both  lots:  if  A’s  wheat  is 
worth  20  %  more  than  B’s,  how  much  ought  I  to  pay  each  ? 

161.  Two  men  divided  a  lot  of  wood,  which  they  pur¬ 
chased  together  for  $27;  one  took  5^-  cords,  the  other  8: 
what  ought  each  to  pay? 

162.  If  8  men  cut  84  cords  of  wood  in  12  days,  working 
7  hours  a  day,  how  many  men  will  cut  150  cords  in  10 
days,  working  5  hours  a  day? 

163.  If  16  horses  consume  84  bushels  of  grain  in  24  days, 
how  many  bushels  of  grain  will  supply  36  horses  16  days? 

164.  If  the  wages  of  24  men  for  4  days  are  $192,  what 
will  be  the  wages  of  36  men  for  3  days? 

165.  If  4  men  in  7f  days  earn  $53J,  how  much  will  7 
men  earn  in  of  a  day  ? 

166.  A  and  B  traded  in  company  and  gained  $750,  of 
which  B’s  share  was  $600;  A’s  stock  was  $1200:  what  was 
B’s  stock? 

167.  A  and  B  formed  a  partnership  for  1  year,  and  A 
put  in  $2000  and  B  $800:  how  much  more  must  B  put  in 
at  the  close  of  6  months  to  receive  one-half  of  the  profits  ? 

168.  A  and  B  engage  in  trade ;  A  puts  in  $200  for  5 
months,  B  $300  for  2  months ;  they  draw  out  capital  and 
profits  to  the  amount  of  $1389 :  what  was  each  man’s  share? 

169.  What  is  the  square  root  of  41616?  Of  420.25? 

170.  What  is  the  cube  root  of  46656?  Of  42.875? 

171.  A  certain  window  is  30  feet  from  the  ground:  how 
far  from  the  base  of  the  building  must  the  foot  of  a  ladder 
50  feet  long  be  placed  to  reach  the  window? 


GENERAL  REVIEW. 


279 


172.  Two  men  start  from  the  same  point ;  one  travels 
52  miles  north  and  the  other  39  miles  west :  how  far  are 
they  apart? 

173.  A  house  is  40  feet  high  from  the  ground  to  the  eaves, 
and  it  is  required  to  find  the  length  of  a  ladder  which  will 
reach  the  eaves,  supposing  the  foot  of  the  ladder  can  not  be 
placed  nearer  to  the  house  than  30  feet. 

174.  How  many  rods  of  fence  will  inclose  10  acres  in  the 
form  of  a  square  ? 

175.  A  floor  is  24  feet  long  and  15  feet  wide:  what  is 
the  distance  between  two  opposite  corners? 

176.  A  room  is  20  feet  long,  16  feet  wide,  and  12  feet 
high :  what  is  the  distance  from  one  of  the  lower  corners  to 
the  upper  opposite  corner? 

177.  How  many  cubic  feet  in  a  stone  8  feet  long,  5^-  feet 
wide,  and  3^  feet  thick? 

178.  How  many  square  feet  on  the  surface  of  a  stone  6 
feet  long,  4  feet  wide,  and  1^  feet  thick? 

179.  There  is  a  circular  field  40  rods  in  diameter:  what 
is  its  circumference?  How  many  acres  does  it  contain? 

180.  The  area  of  a  circle  is  470.8f  square  inches:  what 
is  the  length  of  its  diameter? 

181.  How  many  iron  balls  2  inches  in  diameter,  will 
weigh  as  much  as  an  iron  ball  8  inches  in  diameter? 

182.  How  many  cubical  blocks,  each  edge  of  which  is  ^ 
of  a  foot,  are  equivalent  to  a  block  of  wood  8  feet  long,  4 
feet  wfide,  and  2  feet  thick? 

183.  How  many  bushels  of  wheat  will  fill  a  bin  8  feet 
long,  5  feet  wide,  and  4  feet  deep? 

184.  How  many  gallons  of  water  will  a  cistern  contain 
which  is  7  feet  long,  6  feet  wide,  and  11  feet  deep? 

185.  How  many  gallons  of  water  will  fill  a  round  cistern 
6  feet  deep  and  4  feet  in  diameter? 

186.  Divide  $1000  among  A,  B,  and  C,  and  give  A  $120 
more  than  C,  and  C  $95  more  than  B. 

187.  A  can  mow  2  acres  in  3  days,  and  B  5  acres  in  6 
days:  in  how  many  days  can  they  together  mow  9  acres? 


280 


COMPLETE  ARITHMETIC. 


188.  A  sold  cloth  to  B  and  gained  10  per  cent ;  B  sold 
it  to  C  and  gained  10  per  cent;  C  sold  it  to  D  for  $726 
and  gained  1 0  per  cent :  how  much  did  it  cost  A  ? 

189.  A  man  steps  2  feet  8  inches,  and  a  boy  1  foot  10 
inches;  but  the  boy  takes  8  steps  while  the  man  takes  5: 
how  far  will  the  boy  walk  while  the  man  walks  3f  miles  ? 

190.  A  father  bequeathed  of  his  estate  to  his  eldest 
son,  §  of  the  remainder  to  his  second  son,  and  the  rest  to 
his  youngest  son ;  by  this  arrangement  the  eldest’ s  share 
was  $1300  more  than  the  youngest’s:  what  was  the  share 
of  each  son  ? 

191.  If  7  bushels  of  wheat  are  worth  10  bushels  of  rye, 
and  5  bushels  of  rye  are  worth  14  bushels  of  oats,  and 
6  bushels  of  oats  are  worth  $3.13,  how  many  bushels  of 
wheat  will  $50  buy? 

192.  In  a  company  of  90  persons  there  are  4  more  men 
than  women  and  10  more  children  than  men  and  women 
together:  how  many  of  each  in  the  company? 

Suggestion:  ^  of  (90 — 10)=  number  of  men  and  women. 

193.  Divide  $630  among  3  persons,  so  that  the  second 
shall  have  §  as  much  as  the  first,  and  the  third  ^  as  much 
as  the  other  two  together.  [Sug.  :  Third’s  share  =  £  of  $630.] 

194.  A  and  B  can  do  a  piece  of  work  in  12  days,  B  and 
C  in  9  days,  and  A  and  C  in  6  days :  how  long  will  it  take 
each  alone  to  do  it? 

195.  A  and  B  perform  together  T9^  of  a  piece  of  work 
in  2  days,  when,  B  leaving,  A  completes  it  in  J  a  day: 
in  what  time  can  each  do  it  alone? 

196.  C  and  D  engage  in  trade  with  different  sums  of 
money;  C  loses  40  per  cent  of  his  capital,  and  D  gains  50 
per  cent  on  his,  when  their  capitals  are  equal:  how  much 
greater  was  C’s  capital  than  D’s  when  they  began  business? 

*  197.  A  man  walked  100  miles  in  2  days,  and  J  of  the 
distance  walked  the  first  day  equaled  \  the  distance  walked 
the  second  day:  how  far  did  he  walk  each  day? 

198.  How  far  from  the  end  of  a  stick  of  timber  30  feet 
long,  of  equal  size  from  end  to  end,  must  a  handspike  be 


TEST  QUESTIONS. 


281 


placed  so  that  3  men,  2  at  the  handspike  and  1  at  the  end 
of  the  stick,  may  each  carry  J  of  its  weight  ? 

Suggestion. — Since  the  handspike  is  to  support  two  thirds  of  the 
weight,  or  twice  as  much  as  is  carried  by  the  man  at  the  end  of  the 
stick,  it  must  be  placed  half  as  far  from  the  middle  of  the  stick,  which 
is  half  the  distance  from  the  end  to  the  middle;  \  of  15  ft.  is  7^  ft.  The 
weights  sustained  at  the  two  points  of  support,  are  inversely  as  their 
respective  distances  from  the  middle  of  the  stick;  2:1  :  :  15  ft  :  Ans. 

199.  Two  trees  stand  on  opposite  sides  of  a  stream  40  feet 
wide ;  the  height  of  one  tree  is  to  the  width  of  the  stream 
as  8  is  to  4,  and  the  width  of  the  stream  is  to  the  height  of  the 
other  as  4  is  to  5 ;  what  is  the  distance  between  their  tops  ? 

Suggestion. — Base  of  right-angled  triangle  is  40  feet,  and  its  per¬ 
pendicular  30  feet. 

200.  A  cistern  is  filled  by  two  pipes,  one  of  which  will 
fill  it  in  2  hours,  and  the  other  in  3  hours;  it  is  emptied 
by  three  pipes,  the  first  of  which  will  empty  it  in  5  hours, 
the  second  in  6  hours,  and  the  third  in  7tt  hours:  if  all  the 
pipes  be  left  open,  in  what  time  will  it  be  filled? 

TEST  QUESTIONS. 

1.  What  is  a  number?  In  how  many  ways  may  numbers  be  rep¬ 
resented?  Name  them. 

2.  What  is  the  difference  between  numeration  and  notation? 
Between  the  Arabic  notation  and  the  Roman? 

3.  What  is  the  value  of  the  figure  5  in  452?  How  is  the  value 
of  a  figure  affected  by  its  removal  one  order  to  the  left?  One 
order  to  the  right?  How  is  the  value  of  a  number  affected  by  an¬ 
nexing  a  cipher?  Why? 

4.  How  many  units  are  there  in  the  sum  of  two  or  more  integers? 
Why.  in  addition,  are  like  orders  of  figures  written  in  the  same 
column?  Why,  in  adding  numbers,  do  we  begin  at  the  right  hand? 

5.  Why  are  the  minuend,  subtrahend,  and  difference  like  num¬ 
bers?  Show  that  the  adding  of  10  to  a  term  of  the  minuend,  and  1 
to  the  next  higher  term  of  the  subtrahend,  increases  the  minuend 
and  subtrahend  equally. 

6.  Why  must  the  multiplier  be  an  abstract  number?  When  the 
multiplicand  is  concrete,  what  is  true  of  the  product?  Why? 

7.  What  kind  of  number  is  the  quotient  when  both  divisor  and 
dividend  are  like  numbers?  What  is  the  difference  between  short 

division  and  long  division? 

C.  Ar.— 24. 


282 


COMPLETE  ARITHMETIC. 


8.  How  is  the  quotient  affected  by  multiplying  or  dividing  both 
dividend  and  divisor  by  the  same  number  ?  By  multiplying  the  divi¬ 
dend  by  any  number  greater  than  1  ?  On  what  principle  may  the  four 
fundamental  rules  be  reduced  to  two  ? 

9.  Name  all  the  prime  numbers  from  1  to  20  inclusive.  Show  that 
two  composite  numbers  may  be  prime  with  respect  to  each  other. 
What  is  meant  by  the  factors  of  a  number?  The  prime  factors? 
Show  that  the  common  factor  of  two  or  more  numbers  is  a  factor  of 
their  sum. 

10.  Why  is  the  factor  of  a  number  its  divisor  ?  How  is  a  number 
affected  by  the  canceling  of  a  factor?  On  what  principle  may  the 
common  factors  of  a  dividend  and  a  divisor  be  canceled  ? 

11.  When  is  a  divisor  a  common  divisor?  What  is  the  greatest 
common  divisor  of  two  or  more  numbers?  Show  that  the  common 
divisor  of  two  numbers  is  a  divisor  of  their  sum  and  difference.  In 
how  many  ways  may  the  greatest  common  divisor  of  two  or  more 
numbers  be  found  ? 

12.  How  many  multiples  has  every  number?  What  is  a  common 
multiple?  What  is  the  least  common  multiple  of  two  or  more 
numbers?  In  how  many  ways  may  the  least  common  multiple  of 
two  or  more  numbers  be  found  ? 

13.  What  is  the  difference  between  a  divisor  and  a  multiple  of  a 
number  ?  Between  the  terms  factor,  divisor,  and  measure  ?  Is  2 J  a 
divisor  of  5?  Is  a  multiple  of  5?  Is  12£  a  multiple  of  61? 

14.  What  is  a  fraction?  In  what  two  ways  may  a  fraction  be  ex¬ 
pressed?  When  a  fraction  is  expressed  by  words,  which  word  or 
words  denote  the  denominator? 

15.  What  is  the  difference  between  the  unit  of  a  fraction  and  o  frac¬ 
tional  unit  ?  Which  term  of  a  fraction  denotes  the  size  of  the  fractional 
unit?  When  is  the  value  of  a  fraction  equal  to  1?  Greater  than  1? 
Less  than  1  ? 

16.  Show  that  the  division  or  multiplication  of  both  terms  of  a 
fraction  by  the  same  number,  does  not  change  its  value.  How  is  the 
value  of  a  proper  fraction  affected  by  adding  the  same  number  to  both 
of  its  terms?  By  subtracting  the  same  number  from  both  of  its  terms? 

17.  On  what  principle  is  a  fraction  reduced  to  lower  terms?  To 
higher  terms?  On  what  principle  are  two  or  more  fractions  reduced 
to  a  common  denominator? 

18.  In  what  two  ways  may  a  fraction  be  multiplied  by  an  integer? 
Why?  In  what  two  ways  may  a  fraction  be  divided  by  an  integer? 
In  what  three  ways  may  a  fraction  be  divided  by  a  fraction?  Why 
must  fractions  have  a  common  denominator  before  they  can  be  added 
or  subtracted? 


TEST  QUESTIONS. 


283 


19.  What  is  a  decimal  fraction  ?  Is  the  fraction  fifteen-hundredths  a 
decimal  fraction?  In  what  two  ways  may  it  be  expressed  by  figures? 
Which  is  called  the  decimal  form?  What  is  the  denominator  of  a 
decimal  fraction? 

20.  What  is  meant  by  decimal  places?  What  is  the  name  of  the 
third  decimal  order  from  units?  The  sixth?  The  ninth?  How  is 
a  decimal  read  ? 

21.  How  is  the  local  value  of  a  decimal  figure  affected  by  its  re¬ 
moval  one  order  to  the  right  ?  One  order  to  the  left  ?  How  is  the 
value  of  a  decimal  affected  by  annexing  decimal  ciphers  ?  Why  ? 
By  prefixing  decimal  ciphers?  Why? 

22.  How  is  a  decimal  reduced  to  a  common  fraction  ?  A  common 
fraction  to  a  decimal?  Why  can  decimals  be  added  and  subtracted 
like  integers? 

23.  Why  does  the  product  contain  as  many  decimal  places  as  both 
multiplicand  and  multiplier?  Why  does  the  dividend  contain  as 
many  decimal  places  as  both  divisor  and  quotient? 

24.  How  is  a  decimal  divided  by  10,  100,  etc.  ?  How  is  a  decimal 
multiplied  by  10,  100,  etc.  ?  Why  are  numbers  denoting  sums  of 
money  added  and  subtracted  like  decimals? 

25.  What  is  a  rectangle  ?  How  is  its  area  found  ?  What  is  a  circle  ? 
How  is  its  area  found? 

26.  What  is  a  right-angled  triangle?  How  is  its  area  found? 

27.  What  is  a  rectangular  solid?  What  is  the  difference  between 
an  edge  and  a  face  of  a  solid? 

28.  Show  that  the  product  of  the  three  dimensions  of  a  rectangular 
solid  represents  its  volume  or  solid  contents.  Plow  are  the  contents  of 
a  cylinder  found? 

29.  Is  every  concrete  number  denominate?  Give  examples.  What 
is  the  difference  between  a  simple  denominate  number  and  a  compound 
number?  Give  examples. 

30.  What  do  denominate  numbers  express?  What  is  the  difference 
between  reduction  descending  and  reduction  ascending? 

31.  How  are  denominate  fractions  reduced  from  a  higher  to  a  lower 
denomination  ?  From  a  lower  to  a  higher  ?  How  is  a  denominate 
number  reduced  to  the  fraction  of  a  higher  denomination?  Give  an 
example. 

32.  What  is  the  Metric  System  ?  What  is  the  primary  unit  of  the 
system  ?  What  is  its  length  in  inches  ?  What  is  a  liter  ?  What  is  a 
gram. 

33.  How  are  the  multiples  of  the  meter,  liter,  and  gram  named? 
How  are  the  subdivisions  named  ? 

34.  What  is  the  difference  between  simple  addition  and  compound 
addition?  In  what  respect  are  the  processes  alike? 


284 


COMPLETE  ARITHMETIC. 


35.  When  are  compound  numbers  of  the  same  kind  ?  Give  exam¬ 
ples.  How  is  a  compound  number  divided  by  another  of  the  same 
kind  ? 

36.  What  part  of  the  equator  passes  beneath  the  vertical  rays  of  the 
sun  every  hour?  What  part  of  the  tropic  of  Cancer?  What  part  of 
any  parallel  situated  between  the  polar  circles? 

37.  Why  is  the  time  of  day  earlier  at  New  York  than  at  St.  Louis? 
When  the  difference  in  longitude  between  two  places  is  given,  how  is 
kthe  difference  in  time  found? 

38.  What  is  meant  by  5  per  cent  of  a  number?  What  is  the  dif¬ 
ference  between  the  terms  rate  per  cent  and  rate  ?  Give  examples. 

39.  What  four  numbers  are  considered  in  percentage  ?  Define  each. 
Give  the  four  cases  of  percentage  and  the  formula  for  each. 

40.  What  is  the  difference  between  the  cost  and  the  selling  price 
of  an  article?  Give  the  four  formulas  in  profit  and  loss. 

41.  What  is  meant  by  commission?  What  is  the  difference  between 
a  factor  and  a  broker  ?  Give  the  four  formulas  in  commission  and 
brokerage. 

42.  What  is  the  difference  between  the  market  value  and  the  par 
value  of  capital?  When  is  capital  at  a  premium?  When  is  it  at  a 
discount? 

43.  What  is  the  difference  between  a  dividend  and  an  assessment? 
How  is  the  rate  of  dividend  found  ? 

44.  What  is  insurance  ?  What  is  fire  insurance  ?  What  is  the 
premium?  Give  the  formulas  covering  the  four  cases  in  insurance. 

45.  What  is  life  insurance?  How  is  the  premium  computed? 
What  is  a  mutual  insurance  company  ? 

46.  What  is  the  difference  between  a  poll  tax  and  a  property  tax  ? 
How  is  a  property  tax  assessed?  How  is  the  rate  of  tax  determined? 

47.  What  is  an  income  tax?  An  excise  tax?  From  what  kind  of 
taxes  is  the  internal  revenue  of  the  United  States  chiefly  derived? 

48.  What  are  customs  or  duties?  What  is  the  difference  between 
specific  duties  and  ad  valorem  duties?  What  is  a  tariff? 

49.  What  is  interest?  What  is  the  rate  of  interest? 

50.  How  is  the  interest  of  any  principal  for  one  year,  at  any  rate 
per  cent  found  ?  Give  the  formula  for  the  general  method  of  com¬ 
puting  interest.  Give  the  formula  for  the  six  per  cent  method. 

51.  How  many  methods  are  there  of  finding  the  time  between  two 
dates?  Which  is  called  the  method  by  days? 

52.  On  what  principle  is  the  United  States  Rule  for  partial  pay¬ 
ments  based?  What  rule  is  used  when  a  note  runs  less  than  a  year? 

53.  What  quantities  are  considered  in  interest  ?  State  the  five 
problems  in  interest,  and  give  the  formula  for  each. 


TEST  QUESTIONS. 


285 


54.  What  is  discount?  What  is  the  difference  between  true  dis¬ 
count  and  interest?  Between  true  discount  and  bank  discount?  Be¬ 
tween  bank  discount  and  interest? 

55.  What  is  meant  by  days  of  grace?  When  does  a  note  with 
grace  become  due?  How  is  a  note  not  drawing  interest  discounted 
by  a  bank  ?  IIow  is  a  note  drawing  interest  discounted  ? 

56.  What  is  a  promissory  note?  What  is  its  face?  Who  is  an  in¬ 
dorser  ?  When  is  a  note  negotiable?  When  is  a  note  not  negotiable? 

57.  What  is  a  draft  ?  What  are  the  names  of  the  three  parties 
named  in  a  draft  ?  What  is  meant  by  the  acceptance  of  a  draft  ?  By 
its  protest?  What  is  exchange?  The  rate  of  exchange? 

58.  What  is  a  bond?  What  is  a  coupon?  When  bonds  are 
quoted  at  108,  what  are  they  worth?  What  are  United  States 
bonds  also  called? 

59.  What  is  annual  interest  ?  When  annual  interest  is  not  paid 
when  due,  what  kind  of  interest  does  it  draw  until  paid? 

60.  What  is  compound  interest?  In  what  respect  does  compound 
interest  differ  from  annual  interest  ? 

61.  On  what  principle  is  the  common  method  of  finding  the  equated 
time  of  several  debts  or  payments  based  ?  What  is  meant  by  the  equa¬ 
tion  of  accounts  ? 

62.  Define  ratio.  In  how  many  and  what  ways  may  the  ratio  of 
two  numbers  be  expressed  ?  What  are  the  two  terms  of  a  ratio 
called  ?  Which  is  the  dividend  ?  When  is  the  value  of  a  ratio  less 
than  one  ?  When  is  it  greater  than  one  ? 

63.  Why  must  the  two  terms  of  a  ratio  be  like  numbers?  Why  is 
the  value  of  a  ratio  not  changed  by  multiplying  or  dividing  both  of 
its  terms  by  the  same  numbers? 

64.  What  is  a  compound  ratio  ?  How  is  a  compound  ratio  reduced 
to  a  simple  ratio? 

65.  What  is  a  proportion  ?  How  many  ratios  in  a  simple  propor¬ 
tion  ?  When  is  a  proportion  called  simple  ?  When  is  it  compound  ? 
How  many  terms  in  a  simple  proportion  ? 

66.  Which  terms  are  called  the  extremes,  and  which  the  means  ? 
To  what  is  the  product  of  the  extremes  equal? 

67.  How  can  a  missing  mean  be  found  ?  Why  ?  A  missing  ex¬ 
treme?  Why?  If  the  second  term  of  a  proportion  is  greater  than 
the  first  term,  how  will  the  fourth  term  compare  with  the  third? 

68.  In  stating  a  problem  in  proportion,  which  number  is  made  the 
third  term?  Why?  What  is  the  relation  between  the  ratio  of  like 
causes  and  the  ratio  of  their  effects  ? 

69.  How  may  a  compound  proportion  be  reduced  to  a  simple  propor¬ 
tion?  How  may  the  fourth  term  of  a  compound  proportion  be  found? 


286 


COMPLETE  ARITHMETIC. 


70.  What  is  the  difference  between  a  simple  partnership  and  a  com¬ 
pound  partnership  ?  On  what  does  the  partnership  value  of  capital 
depend  ? 

71.  What  is  the  difference  between  the  power  of  a  number  and  its 
root?  Give  examples.  What  is  the  difference  between  involution 
and  evolution? 

72.  What  is  the  difference  between  a  perfect  power  and  an  imper¬ 
fect  power?  Give  examples.  When  is  a  root  called  a  surd  ? 

73.  To  what  is  the  square  of  a  number  composed  of  tens  and  units 
equal  ?  To  what  is  the  cube  of  a  number  composed  of  tens  and  units 
equal  ? 

74.  How  many  orders  in  the  square  of  any  number?  How  many 
orders  in  the  square  root  of  any  number?  How  many  orders  in  the 
cube  of  any  number?  How  many  orders  in  the  cube  root  of  any 
number  ? 

75.  How  is  the  first  term  of  the  square  root  of  any  number  found? 
The  second  term?  How  is  the  first  term  of  the  cube  root  of  any 
number  found?  The  second  term ? 

76.  To  what  is  the  square  of  the  hypotenuse  of  a  right-angled  tri¬ 
angle  equal  ?  The  square  of  the  base  or  perpendicular  ? 

77.  How  may  the  area  of  a  circle  be  found  ?  When  the  area  is 
given,  how  may  the  diameter  be  found?  What  is  the  relation  be¬ 
tween  the  areas  of  two  circles? 

78.  How  is  the  surface  of  a  sphere  found?  Its  solidity?  What  is 
the  relation  between  the  solid  contents  of  two  spheres? 

Solution  of  Watch  Problems,  page  243. 

The  dial  of  a  watch  is  divided  into  12  equal  spaces,  and  while 
the  hour  hand  passes  over  1  of  these  spaces,  the  minute  hand 
passes  over  12,  and  hence  the  minute  hand  passes  over  12  spaces 
to  gain  11  spaces  on  the  hour  hand,  or  of  a  space  to  gain  1  space. 
But  it  takes  the  minute  hand  5  minutes  to  pass  over  1  space ;  and 
to  gain  1  space  on  the  hour  hand,  it  will  take  j-f-  of  5  minutes,  or 
5t5t  minutes.  Hence,  it  will  take  5T5T  minutes  for  the  minute  hand 
to  overtake  the  hour  hand  between  1  and  2  o’clock  (to  gain  1  space), 
twice  5x5t  minutes  between  2  and  3  o’clock,  3  times  5T5T  minutes  be¬ 
tween  3  and  4  o’clock,  etc. 


APPENDIX. 


NOTATION. 

423.  In  the  decimal  system  of  notation,  with  ten  for  its 
base,  ten  figures  are  used ;  in  a  system  with  twenty  for  its 
base,  twenty  figures  would  be  needed ;  in  a  system  with  five 
for  its  base,  only  five  figures  (1,  2,  3,  4,  0)  would  be  needed ; 
and,  generally,  a  system  of  notation  requires  as  many  different 
figures  as  there  are  units  in  its  base. 

424.  In  a  system  with  five  for  its  base,  24  would  express 
fourteen;  124  would  express  thirty-nine;  1120  would  express 
one  hundred  and  sixty. 


EXERCISES. 

1.  What  number  is  expressed  by  200  on  a  scale  of  five? 

2.  What  number  is  expressed  by  1240  on  a  scale  of  five? 

3.  Express  forty  on  a  scale  of  five, 

4.  Express  one  hundred  on  a  scale  of  five. 

5.  Express  two  hundred  on  a  scale  of  five. 

PROOF  OF  THE  SIMPLE  RULES  BY  “CASTING 

OUT  THE  9’s.” 

425.  The  method  of  proving  the  elementary  operations 
of  arithmetic  by  “casting  out  the  9’s”  is  based  on  the 
principle,  that  the  excess  of  9’s  in  any  number  is  equal  to  the 
excess  of  9’s  in  the  sum  of  its  digits. 

Take,  for  example,  2345.  Dividing  it  by  9,  we  have  the 
remainder  5,  for  the  excess  of  9’s ;  and  adding  the  digits 
(2  3  -f-  4  -j-  5  =  14),  and  dividing  the  sum  by  9,  we  have 

the  same  remainder. 


(287) 


288 


COMPLETE  ARITHMETIC. 


426.  This  principle  may  be  thus  explained : 


2345  4 


2000  =  222  X  9  +  2 
300  —  33  X  9  +  3 
40  =  4X9  +  4 

5  =  5 


It  is  seen  that  2000  is  222  times  9,  with  a  remainder  2 ; 
300  is  33  times  9,  with  a  remainder  3 ;  40  is  4  times  9,  with 
a  remainder  4.  Hence,  the  remainders  obtained  by  dividing 
the  several  parts  of  a  number,  denoted  by  the  local  value 
of  its  digits,  by  9,  are  respectively  the  digits  of  the  number; 
and  the  remainder  obtained  by  dividing  the  number  itself 
by  9,  equals  the  remainder  obtained  by  dividing  the  sum 
of  its  digits  by  9.  Hence, 

The  excess  of  9’s  in  any  number  is  found  by  adding  its  digits 
and  finding  the  excess  of  9’s  in  their  sum. 


427.  Proof  of  Addition. 

The  excess  of  9’s  in  the  first  number,  found  by 
adding  its  digits,  is  1 ;  in  the  second  number,  4 ;  in 
the  third,  7.  The  excess  of  9’s  in  the  sum  of  these 
excesses  is  3,  which  equals  the  excess  of  9’s  in  939, 
the  amount.  Hence, 

The  excess  of  9’s  in  the  sum  of  several  numbers  is 
equal  to  the  excess  of  9’s  in  the  sum  of  their  excesses. 

1.  Add  and  prove  2346,  5084,  6784,  8653,  and  9045. 

2.  Add  and  prove  30483,  50678,  346864,  and  706037. 

3.  Add  and  prove  530902,  672084,  567084,  and  1345602. 


Process. 

325  Excess  1 

256  “  4 

358  “  7 

939  “  3 


Process. 

3676  Excess  4 

1508  “  5 

2168  “  8 

“  4 


428.  Proof  of  Subtraction. 

Since  the  minuend  is  equal  to  the  sum  of  the  sub¬ 
trahend  and  remainder,  the  excess  of  9’s  in  the  minuend 
equals  the  excess  of  9’s  in  the  sum  of  the  excesses  in  the 
subtrahend  and  remainder. 


1.  From  40603  take  27475,  and  prove  the  result. 

2.  From  607853  take  492097,  and  prove  the  result. 


CIRCULATING  DECIMALS. 


289 


429.  Proof  of  Multiplication. 


Since  347  contains  a  certain  number  of  9’s  with 
an  excess  of  5,  and  53  contains  a  certain  number 
of  9’s  with  an  excess  of  8,  the  product  of  347  and 
53  consists  of  the  product  of  the  number  of  9’s 
in  them,  plus  the  product  of  5  and  8,  the  excesses 
of  9’s.  Hence, 

The  excess  of  9’s  in  the  product  of  two  numbers  is 
equal  to  the  excess  of  9’s  in  the  product  of  the  excesses  in  these  numbers. 


Process. 

347  Excess  5 
53  “  8 

40 


1041 
1735 

18391  Excess  4 


1.  Multiply  45603  by  708,  and  prove  the  result. 

2.  Multiply  60875  by  690,  and  prove  the  result. 


430.  Proof  of  Division. 


Process. 

347 )  18496  (  53 
1735 

1146 

1041 

105 

18496  Excess  J. 

347  “  5 

53  “  8 

105  “  6 

5X8  +  6  =  46  “  1 


Since  the  dividend  equals  the  product 
of  divisor  and  quotient,  plus  the  re¬ 
mainder,  the  excess  of  9’s  in  the  divi¬ 
dend  is  equal  to  the  excess  of  9’s  in  the 
product  of  divisor  and  quotient,  plus  the 
excess  in  the  remainder.  Hence, 

The  excess  of  9’s  in  the  dividend  is  equal 
to  the  excess  of  9’s  in  the  product  of  the 
excesses  in  divisor  and  quotient ,  plus  the 
excess  in  the  remainder. 


1.  Divide  6480  by  47,  and  prove  the  result. 

2.  Divide  15685  by  625,  and  prove  the  result. 


CIRCULATING  DECIMALS. 

431.  A  Circulating  Decimal  is  an  interminate 
decimal,  containing  the  same  figure  or  set  of  figures, 
repeated  in  the  same  order  indefinitely.  (Art.  121.) 

432.  The  figure  or  set  of  figures  repeated  is  called  a 
Repetend. 

A  repetend  is  denoted  by  a  dot  placed  over  the  first  and  • 

•  •  •  •  • 

last  of  its  figures;  as,  .5  .16  .325. 

C.Ar.— 25. 


290 


COMPLETE  ARITHMETIC. 


433.  When  a  circulating  decimal  has  no  figure  but  the 

•  • 

repetend,  it  is  called  a  Pure  Circulate ;  as,  .325. 

When  a  circulating  decimal  has  one  or  more  figures 
before  the  repetend,  it  is  called  a  Mixed  Circulate,  as, 
.4526. 

434.  A  pure  circulate  is  reduced  to  a  common  fraction  by 
taking  the  repetend  for  the  numerator,  and  as  many  9’s  for  the 
denominator  as  there  are  figures  in  the  repetend. 

Proof. 

•  • 

Let  .63  be  a  pure  circulate. 


Then, 

63.63 

=  100  times  the 

pure 

circulate. 

.63 

—  1  time  “ 

u 

u 

Subtracting, 

63. 

=  99  times  11 

u 

« 

Hence, 

63 

99 

=  the  value  of 

u 

« 

435.  A  mixed  circulate  is  reduced  to  a  common  fraction 
by  subtracting  the  terms  which  precede  the  repetend  from  the  whole 
circulate,  and  taking  the  difference  for  the  numerator ;  and,  for 
the  denominator,  taking  as  many  9’s  as  there  are  figures  in  the 
repetend ,  with  as  many  ciphers  annexed  as  there  are  decimal 
figures  before  the  repetend. 

Proof. 

•  • 

Let  .45124  be  a  mixed  circulate. 

Then,  45124.i24  =  100000  times  the  mixed  circulate. 

And,  45.i24  =  100  “  “  “  “ 

Subtracting,  45079  =  99900  “  “  “  “ 

Hence,  =  the  value  of  “  “  “ 

436  Pure  or  mixed  circulates  may  be  added,  subtracted, 
multiplied,  or  divided  by  first  reducing  them  to  common 
fractions. 

Note. — Circulates  may  be  added,  subtracted,  multiplied,  or  divided 
without  first  reducing  them  to  common  fractions ;  but  the  processes  are 
not  of  sufficient  practical  importance  to  justify  their  explanation  in  a 
school  arithmetic.  In  all  computations,  circulates  are  carried  to  enough 
places  to  avoid  any  appreciable  error  in  the  result,  and  then  are  treated 
as  other  decimals. 


TABLES. 


291 


437.  TABLES  OF  DENOMINATE  NUMBERS. 

I.  CURRENCIES. 


1.  United  States  Money. 

2.  English  Money. 

The  denominations  are 
mills ,  cents,  dimes ,  dollars, 
and  eagles. 

The  denominations  are  far¬ 
things  (q.),  pence  (d.),  shil¬ 
lings  (s.),  and  pounds  (£). 

Table. 

Table. 

10  m.  =  1  ct. 

10  ct.  =  1  d. 

10  d .  =  $1 
$10  =  1  E. 

4  q.  =  1  d. 

12  d.  =  1  s 

20  s.  =  1  £. 

1  £  =  $4.8665. 

II.  MEASURES  OF  EXTENSION  AND  TIME. 

1.  MEASURES  OF  LINES  AND  ARCS. 


Long  Measure. 

Circular  Measure. 

The  denominations  are 
inches,  feet,  yards,  rods ,  fur¬ 
longs,  and  miles. 

The  denominations  are  sec¬ 
onds,  minutes,  degrees,  signs, 
and  circumferences. 

Table. 

12  in.  =  1  ft. 

3  ft.  =1  yd. 
yd.  =  1  rd. 

40  rd.  =  1  fur. 

8  fur.  =  1  m. 

Table. 

60"  =  V 

60'  =  1° 

30°  =  1  s. 

12 s-  UlC. 

360°  / 

Also : 

Cloth  Measure. 

3  barleycorns  =  1  inch. 

4  inches  =  1  hand. 

3  feet  =  1  pace. 

6  feet  —  1  fathom. 

3  miles  (geog.)—  1  league. 

( Little  used.) 

2\  in.  —  1  nail. 

4  n.  =1  quarter. 

4  qr.  =  1  yard. 

5  qr.  =  1  Ell  Eng. 

60  geographic  miles  ) 

~nl  .  ,  .  • i  /  n  >  =  1  degree  at  the  equator . 

69  J  statute  miles  ( nearly )  J  J  1 


292 


COMPLETE  ARITHMETIC. 


2.  MEASURES  OF  SURFACES  OR  AREAS. 


Square  Measure. 

The  denominations  are 
square  inches,  square  feet, 
square  yards,  square  rods  (or 
perches ),  roods,  acres,  and 
square  miles. 

Table. 

144  sq.  in.  =  1  sq.  ft. 

9  sq.  ft.  =  1  sq.  yd. 

304  sq.  yd.  =  1  P. 

40  P.  =1  B. 

4  B.  =11 
640  A.  =1  sq.  mi. 


Surveyor’s  Measure. 
Table. 

7.92  in.  =  1  link  ( l .). 

25  l.  =1  rod. 

4  rd.  =  1  chain  ( ch.) 
80  ch.  =  1  mile. 

Also : 

625  sq.  1.  =1  P. 

16  P.  =1  sq.  ch. 

10  sq.  ch.  =  1  A. 

640  A.  =1  sq.  mi. 

1  sq.  mi.  =  1  section. 

36  sect.  =  1  township. 


3.  MEASURES  OF  SOLID  CONTENTS  OR  CAPACITY. 


Cubic  Measure. 

The  denominations  are 
cubic  inches ,  cubic  feet,  and 
cubic  yards. 

Table. 

1728  cu.  in.  =  1  cu.  ft. 

27  cu.  ft.  —  1  cu.  yd. 


Dry  Measure. 

The  denominations  are 
pints,  quarts,  pecks ,  and 
bushels. 

Table. 

2  pt.  =1  qt. 

8  qt.  =1  pk. 

4  pk.  1  bu. 


Wood  Measure. 
Table. 

16  cu.  ft.  =  1  cord  ft. 
8  cd.fi.,  or), 

128  cu.  ft.  i 


24f  cu.  ft.  —  1  perch  of  stone. 


Liquid  Measure. 

Table. 

4  gills  =  1  pt. 

2  pt.  =1  qt. 

4  qt.  =1  gal. 


Notes. — 1.  The  standard  bushel  contains  2150f  cu.  in.;  the  liquid 
gallon,  231  cu.  in.;  and  the  beer  gallon  (little  used),  282  cu.  in. 

2.  The  size  of  casks  for  liquids  are  variable.  Barrels  contain 
from  30  to  40  gallons,  or  more.  The  capacity  of  vats,  cisterns,  etc., 
is  usually  measured  in  barrels  of  31 4  gallons. 


TABLES. 


293 


4.  MEASURES  OF  DURATION  OR  TIME. 

Time  Measure. 

The  denominations  are  seconds,  minutes,  hours,  days,  years. 


and  centuries. 

Table. 

60  sec.  =  1  min. 

60  min.  —  1  h. 

24  h.  =1  da. 

365  da.  =  1  common  yr. 

366  da.  =  1  leap  yr. 
365^  da.  =  1  solar  yr. 
100  s.  yr.  —  1  century. 

Also : 

7  da.  =  1  week. 

4  w.  =1  lunar  mo. 


Calendar  Months. 


January, 

1st 

mo., 

31  days. 

February, 

2d 

<« 

28 

or  29. 

March, 

3d 

« 

31  days. 

April, 

4th 

u 

30 

U 

May, 

5th 

u 

31 

a 

June, 

6th 

u 

30 

u 

July, 

7th 

a 

31 

a 

August, 

8th 

« 

31 

u 

September, 

9th 

u 

30 

a 

October, 

10th 

a 

31 

u 

November, 

11th 

u 

30 

u 

December, 

12th 

u 

31 

ii 

Also: 

A  Julian  year  contains  13  lunar  mo.  1  da.  6  h. 

A  civil  year  contains  12  calendar  months. 

A  solar  year  contains  365  da.  5  h.  48  min.  48,  sec. 


III.  WEIGHTS. 


Avoirdupois  Weight. 


The  denominations 
weights,  and  tons. 


are  drams, 
Table. 


ounces ,  pounds,  hundred- 


16  dr.  —  1  oz. 

16  oz.  =  1  lb. 

100  lb.  =  1  cwt. 

20  cwt.  —  IT. 

Also : 

196  lb.  flour  —  1  barrel. 

200  lb.  beef  or  pork  =  1  “ 


100  lb.  fish 
56  lb.  corn  or  rye  \ 
60  lb.  wheat 
32  lb.  oats  J 
14  lb.  iron  or  lead 
21^  stones 
8  pigs 


—  1  quintal. 
=  1  bushel. 
=  1  stone. 

= 1  pig- 

=  1  j other . 


294 


COMPLETE  ARITHMETIC. 


Troy  Weight. 

The  denominations  are 
grains ,  permy  weights,  ounces, 
and  pounds. 

Table. 

24  gr.  —  1  pwt. 

20  pwt.  =  1  oz. 

12  oz.  —  1  lb. 

3J  T.  gr.  =  1  carat. 

Note. — In  determining  the  fine¬ 
ness  of  gold,  it  is  considered  as 
composed  of  24  parts,  called  carats, 
and  the  number  of  carats  specified 
is  the  number  of  24ths  of  pure 
gold  which  it  contains.  A  sixteen- 
carat  chain  contains  \\  of  pure 
gold  and  of  alloy. 


Apothecaries  Weight. 

The  denominations  are 
grains,  scruples,  drams,  ounces , 
and  pounds. 

Table. 

20  gr.  =  1  ^ 

3  9=13 
3  3  =  1  § 

12  %  —  1  lb 


Apothecaries  Fluid  Measure. 

60  minims  =  1  dram,  f  3. 

8/3  =  1  ounce,  f  £. 
16  /  3  =  1  pint,  0. 


COMPARISON  OP  WEIGHTS. 


1  lb.  Avoir. 
1  02.  “ 


IjVi  d).  Troy  =  lT3/?  lb  Apoth. 


—  175  n~ 

—  oz 


—  175 

—  1^2 


a 


IV.  MISCELLANEOUS  TABLES. 


12  things 
12  dozen 
12  gross 
20  things 
18  inches 
22  inches  ( nearly ) 


are  1  dozen. 

“  1  gross. 

“  1  great  gross. 
“  1  score. 

“  1  cubit. 

“  1  sacred  cubit. 


Paper. 


24  sheets 
20  quires 
2  reams 
5  bundles 


are  1  quire. 

“  1  ream. 

“  1  bundle. 

“  1  bale. 


Books. 


A  sheet  folded  in 

tt  u 

tt  tt 

tt  u 

tt  a 

u  tt 


2  leaves 

is  called 

a  folio. 

4 

tt 

a  quarto  or  4 to. 

8 

u 

an  octavo ,  or  8 vo. 

12 

u 

a  duodecimo,  or  12wio. 

16 

it 

a  16mo. 

24 

u 

a  24 mo. 

Note. — In  estimating  the  size  of  the  leaves,  as  above,  the  doubl® 
medium  sheet  (23  by  26  inches)  is  taken  as  a  standard. 


LIFE  INSURANCE. 


295 


LEGAL  RATES  OF  INTEREST. 


438.  The  rates  of  interest  fixed  by  law  in  the  several 
states,  are  as  follows : 


NAME  OF  STATE. 

Legal 
Rate  of 

Interest. 

Rate  al¬ 
lowed  by 
Contr’ct. 

Alabama . 

8  % 

8  % 

Arkansas . 

6% 

10% 

California . 

10  % 

Any. 

Colorado . 

10  % 

Any. 

Connecticut . 

6% 

6% 

Delaware . 

6% 

6% 

Florida  . 

GO 

Any. 

Georgia . 

7% 

Any. 

Illinois  . 

6% 

8  % 

Indiana . 

6% 

8  Jo 

Iowa . 

6% 

10% 

Kansas . 

7% 

12% 

Kentucky  . 

6% 

100 

Louisiana . 

5  % 

80 

Maine  . 

6% 

Any. 

Maryland . 

6% 

6% 

Massachusetts . 

6% 

60 

Michigan . 

7  % 

10% 

Minnesota . 

7% 

100 

NAME  OF  STATE. 

Legal 
Rate  of 
Interest. 

Rate  al¬ 
lowed  by 
Contr’ct. 

Mississippi . 

60 

10% 

Missouri  . 

60 

10% 

Nebraska . 

10% 

12% 

Nevada . 

10% 

Any. 

New  Hampshire... 

6% 

6% 

New  Jersey . 

6% 

6% 

New  York . 

6% 

6% 

North  Carolina . 

6% 

8% 

Ohio . 

6% 

8% 

Oregon . 

10% 

12% 

Pennsylvania . 

6% 

6% 

Rhode  Island . 

6% 

Any. 

South  Carolina . 

7% 

Any. 

Tennessee . 

6% 

6% 

Texas . 

8% 

12% 

Vermont . 

6% 

6% 

Virginia . 

6% 

8% 

West  Virginia . 

6% 

0% 

Wisconsin . 

7% 

10% 

LIFE  INSURANCE. 

439.  The  rate  of  premium  in  life  insurance  is  based  on 
the  applicant’s  expectation  of  life ,  as  shown  by  life  statistics 
or  bills  of  mortality. 

The  annual  premium  must  be  such  a  sum  as,  put  at  interest,  will 
amount  to  the  sum  insured  at  the  close  of  the  average  extension  of 
life  beyond  the  applicant’s  age. 

440.  There  are  two  tables  showing  the  Expectation  of 
Life,  called  the  Carlisle  Table  and  the  Wigglesworth  Table. 
The  former  is  based  on  bills  of  mortality  prepared  in  En¬ 
gland,  and  the  latter  is  based  on  the  mortality  in  the  United 
States.  Both  tables  are  in  use  in  this  country. 


296 


COMPLETE  ARITHMETIC. 


441.  The  Expectation  of  Life,  as  shown  by  the  two 
tables,  is  as  follows  : 


AGE. 

EXPECTATION  BY 
C.  TABLE. 

EXPECTATION  BY 
W.  TABLE. 

AGE. 

EXPECTATION  BY 
C.  TABLE. 

EXPECTATION  BY 

W.  TABLE. 

W 

CJ 

◄ 

EXPECTATION  BY 

C.  TABLE. 

EXPECTATION  BY 

W.  TABLE. 

W 

o 

< 

EXPECTATION  BY 

C.  TABLE. 

EXPECTATION  BY 

W.  TABLE. 

0 

38.72 

28.15 

24 

38.59 

32.70 

48 

22.80 

22.27 

72 

8.16 

9.14 

1 

44.68 

36.78 

25 

37.86 

32.33 

49 

21.81 

21.72 

73 

7.72 

8.69 

2 

47.55 

38.74 

26 

37.14 

31.93 

50 

21.11 

21.17 

74 

7.33 

8.25 

3 

49.82 

40.01 

27 

36.41 

31.50 

51 

20.39 

20.61 

75 

7.61 

7.83 

4 

50.76 

40.73 

28 

35.69 

31.08 

52 

19.68 

20.05 

76 

6.49 

7.40 

5 

51.25 

40.88 

29 

35.00 

30.66 

53 

18.97 

19.49 

77 

6.10 

6.99 

6 

51.17 

40.69 

30 

34.34 

30.25 

54 

18.28 

18.92 

78 

6.02 

6.59 

7 

50.80 

40.47 

31 

33.68 

29.83 

55 

17.58 

18.35 

79 

5.80 

6.21 

8 

50.24 

40.14 

32 

33.03 

29.43 

56 

16.89 

17.78 

80 

5.51 

5.85 

9 

49.57 

39.72 

33 

32.36 

29.02 

57 

16.21 

17.20 

81 

5.21 

5.50 

10 

48.82 

39.23 

34 

31.68 

28.62 

58 

15.55 

16.63 

82 

4.93 

5.16 

11 

48.04 

38.64 

35 

31.00 

28.22 

59 

14.92 

19.04 

83 

4.65 

4.87 

12 

47.27 

38.02 

36 

30.32 

27.78 

60 

14.34 

15.45 

84 

4.39 

4.66 

13 

46.51 

37.41 

37 

29.64 

27.34 

61 

13.82 

14.86 

85 

4.12 

4.57 

14 

45.75 

36.79 

38 

28.96 

26.91 

62 

13.31 

14.26 

86 

3.90 

4.21 

15 

45.00 

36.17 

39 

28.28 

26.47 

63 

12.81 

13.66 

87 

3.71 

3.90 

16 

44.27 

35.76 

40 

27.61 

26.04 

64 

12.30 

13.05 

88 

3.59 

3.67 

17 

43.57 

35.37 

41 

26.97 

25.61 

65 

11.79 

12.43 

89 

3.47 

3.56 

18 

42.87 

34.98 

42 

26.34 

25.19 

66 

11.27 

11.96 

90 

3.28 

3.73 

19 

42.17 

34.59 

43 

25.71 

24.77 

67 

10.75 

11.48 

91 

326 

3.32 

20 

41.46 

34.22 

44 

25.09 

24.35 

68 

10.23 

11.01 

92 

3.37 

3.12 

21 

40.75 

33.84 

45 

24.46 

23.92 

69 

9.70 

10  50 

93 

3.48 

2.40 

22 

40.04 

33.46. 

46 

23.82 

23.37 

70 

9.18 

10.06 

94 

3.53 

1.98 

23 

39.31 

33.08 

47 

23.17 

22.83 

71 

8.65 

9.60 

95 

3.53 

1.62 

Note. — A  comparison  shows  that  the  Wigglesworth  table  has  a 
less  expectation  of  life  than  the  Carlisle  table  for  all  ages  below  50 
years ;  and  that  the  latter  table  has  a  less  expectation  than  the  former 
for  all  ages  from  50  to  90  years  inclusive. 


EQUATION  OF  PAYMENTS. 

442.  In  1860,  the  author  published  a  demonstration  of  the 
correctness  of  the  common  Mercantile  Rule  for  finding  the 
equated  time  for  the  payment  of  several  debts,  due  at  dif¬ 
ferent  times  without  interest.  The  inaccuracy  of  the  rule 
by  present  worths ,  commended  by  several  authors  as  “the  only 
accurate  rule  ”,  was  thus  pointed  out : 


ARITHMETICAL  PROGRESSION. 


297 


“  The  equated  time  for  the  payment  of  $200,  of  which  $100  is  now 
due,  and  the  other  $100  is  due  in  two  years,  as  found  by  this  rule,  is 
11.32  months.  Now,  the  amount  of  $100  for  11.32  months,  at  6  per 
cent.,  is  $105.66;  the  present  worth  of  the  other  $100,  due  in  12.679 
months,  is  $94,038,  and  $105.66  -{-  $94,038  =  $199,698,  whereas  it 
ought  to  be  $200. 

“  It  is  also  evident  that  the  equated  time,  as  found  by  this  ‘accurate’ 
rule,  will  not  be  the  same  for  all  rates  of  interest.  At  50  per  cent, 
the  equated  time  of  the  above  example  is  8  months,  and  the  error,  by 
the  above  test,  $8.33£;  at  100  per  cent,  it  is  6  months,  with  an  error 
of  $10. 

“This  supposed  accurate  rule  is  based  upon  the  principle  that  the 
amount  to  be  paid  on  a  debt  due  at  a  future  date,  without  interest,  at 
any  time  'previous  to  this  date ,  is  the  present  worth  of  the  debt  at  any  prior 
date,  plus  the  interest  of  the  present  worth  up  to  date  of  payment. 
The  incorrectness  of  this  principle  is  easily  shown.  Suppose  I  owe  a 
man  $100,  due  in  two  years,  without  interest;  how  much  ought  I  to 
pay  in  one  year? 

“The  present  worth  of  $100,  due  in  two  years  (at  6  per  cent),  is 
$89.2857,  and  the  interest  on  this  sum  for  one  year  is  $5.3571 ;  hence, 
the  sum  to  be  paid  is  $89.2857  -f-  $5.3571  =  $94.6428.  The  true 
amount  to  be  paid,  however,  is  the  present  worth  of  $100,  due  in  one 
year,  which  is  $94,339.” 

Note. — The  accuracy  of  the  Mercantile  Rule  and  the  inaccuracy 
of  the  rule  by  Present  Worths  were  rigidly  demonstrated  by  Prof.  A. 
Schuyler,  in  an  article  published  in  the  Ohio  Educational  Monthly ,  for 

1862,  p.  116. 

ARITHMETICAL  PROGRESSION. 

443.  An  Arithmetical  Progression  is  a  series  of 
numbers  which  so  increases  or  decreases  that  the  difference 
between  the  consecutive  numbers  is  constant. 

444.  The  numbers  which  form  the  series  are  called 
Terms ,  the  first  and  last  terms  being  the  Extremes ,  and  the 
intervening  terms  the  Means. 

The  difference  between  the  consecutive  terms  is  called  the 
Common  Difference. 

445.  An  Ascending  Series  is  one  in  which  the  terms  in¬ 
crease;  as,  2,  5,  8,  11,  14,  etc. 

A  Descending  Series  is  one  in  which  the  terms  decrease ; 
as  20,  17,  14,  11,  8,  etc. 


446.  In  an  arithmetical  progression  five  quantities  are 


298 


COMPLETE  ARITHMETIC. 


considered;  and  such  is  the  relation  between  them,  that,  if 
any  three  are  given,  the  other  two  may  be  found. 

These  quantities  are: 

1.  The  first  term. 

2.  The  last  term. 

8.  The  common  difference. 

4.  The  number  of  terms. 

5.  The  sum  of  all  the  terms. 


447.  The  ascending  series,  2,  5,  8,  11,  14,  having  5  terms, 
may  be  expressed  in  three  forms,  as  follows : 


(1) 

2 

5 

8 

11 

14 

(2) 

2 

24-3 

2-K34-3) 

2+(3+3+3) 

2+(3+3+3+3) 

(3) 

2 

24-3 

2+3X2 

2+3X3 

2+3X4 

A  comparison  of  these  three  forms  of  the  same  series 
shows,  that  each  term  is  composed  of  two  parts,  viz.:  (1)  the 
first  term ;  (2)  the  common  difference  taken  as  many  times 
as  there  are  preceding  terms.  Hence, 

1.  The  last  term  of  an  ascending  series  is  equal  to  the  first 

term,  plus  the  common  difference  taken  as  many  times  as  there 

are  terms  in  the  series  less  one.  Conversely, 

2.  The  first  term  of  an  ascending  series  is  equal  to  the  last 
term,  minus  the  common  difference  taken  as  many  times  as  there 
are  terms  in  the  series  less  one. 

3.  The  common  difference  is  equal  to  the  difference  between 
the  first  and  last  terms,  divided  by  the  number  of  terms  less  one. 

4.  The  number  of  terms  less  one  is  equal  to  the  difference 
between  the  first  and  last  terms,  divided  by  the  common  difference. 

448.  Let 

3  5  7  9  11  13  be  an  arithmetical  series, 

and,  13  11  9  7  5  3  be  the  series  reversed. 


Then,  16  +  16  +  16  -f-  16  +  16  +  16  —  twice  the  sum  of  the  terms, 
and  8-[“  8  +  8  +  8  +  84“  8  =  the  sum  of  the  terms. 


ARITHMETICAL  PROGRESSION. 


299 


An  inspection  of  the  above  shows  that  the  sum  of  the  first 
and  last  terms  of  an  arithmetical  series,  multiplied  by  the 
number  of  terms,  is  equal  to  twice  the  sum  of  all  the  terms. 
Hence,  The  sum  of  all  the  terms  of  an  arithmetical  series  is 
equal  to  the  'product  of  one  half  the  sum  of  the  first  and  last 
terms,  multiplied  by  the  number  of  terms. 


Note. — One  half  of  the  sum  of  the  first  and  last  terms  is  equal  to 
the  average  of  the  several  terms  of  the  series. 

449.  From  the  above  principles  may  be  deduced  the  fol¬ 
lowing 


FORMULAS. 


1.  Last  term  =  first  term  ±  ( com .  difference  X  number  of 
terms  less  one). 

2.  First  term  =  last  term  (com.  difference  X  number  of 
terms  less  one). 

3.  Common  difference  =  \  Ud  term  ~first  Urm  j  -4-  number 

( first  term  —  last  term  j 

of  terms  less  one. 

.  7  -  .  7  f  last  term  —  first  term ) 

4.  dumber  of  terms  less  one  =  ■>  ^  term  ] 

— j—  common  difference. 

5.  Sum  of  terms  =  \  ( first  term  -J-  last  term)  X  number  of 
terms. 


Note. — The  first  term  of  an  ascending  series  corresponds  to  the  last 
term  of  a  like  descending  series,  and  the  last  term  of  a  descending  series 
corresponds  to  the  first  term  of  a  like  ascending  series. 


PROBLEMS. 

1.  What  is  the  tenth  term  of  the  series  5,  7,  9,  11,  etc.? 

2.  The  first  term  of  an  ascending  series  is  4,  the  common 
difference  3,  and  the  number  of  terms  8 :  what  is  the  last 
term  ? 

3.  The  last  term  of  a  descending  series  is  1,  the  common 
difference  4,  and  the  number  of  terms  12 :  what  is  the  first 
term  ? 


300 


COMPLETE  ARITHMETIC. 


4.  The  extremes  of  an  arithmetical  series  are  47  and  3, 
and  the  number  of  terms  12:  what  is  the  common  differ¬ 
ence  ? 

5.  The  1st  term  is  7  and  the  21st  term  57 :  what  is  the 
common  difference? 

6.  The  4th  term  of  a  series  is  21  and  the  9th  term  is  41 : 
what  are  the  four  mean  terms? 

7.  The  two  extremes  of  a  series  are  12  and  177,  and  the 
common  difference  5:  what  is  the  number  of  terms? 

8.  The  two  extremes  of  a  series  are  20  and  152,  and  the 
number  of  terms  45 :  what  is  the  sum  of  all  the  terms  ? 

9.  What  is  the  sum  of  all  the  terms  of  the  series  described 
in  the  6th  problem  above?  In  the  7th  problem? 

10.  How  many  strokes  does*  the  hammer  of  a  clock  make 
in  24  hours? 

11.  A  man  agreed  to  dig  a  trench  50  yards  long  for  2 
cents  for  the  first  yard,  5  cents  for  the  second  yard,  8  cents 
for  the  third,  and  so  on,  the  price  of  each  yard  being  3 
cents  more  than  that  of  the  preceding  yard :  what  did  he 
receive  for  digging  the  last  yard?  For  digging  the  trench? 


GEOMETRICAL  PROGRESSION. 

450.  A  Geometrical  Progression  is  a  series  of 
numbers  which  so  increases  or  decreases  that  the  ratio  be¬ 
tween  the  consecutive  terms  is  constant. 

The  first  and  last  terms  are  called  the  Extremes ,  and  the 
intervening  terms  are  called  the  Means . 

451.  A  geometrical  progression  is  ascending  or  descending 
according  as  the  series  increases  or  decreases  from  left  to 
right. 

452.  In  a  geometrical  progression  five  quantities  are  con¬ 
sidered,  and  these  (as  in  arithmetical  progression)  are  so 
related  to  each  other  that,  any  three  being  given,  the  other 
two  may  be  found. 


GEOMETRICAL  PROGRESSION. 


301 


These  five  quantities  are 

1.  The  first  term. 

2.  The  last  term. 

3.  The  common  ratio. 

4.  The  number  of  terms. 

5.  The  sum  of  all  the  terms. 

453.  The  ascending  series,  2,  6,  18,  54,  162,  486,  has  6 
terms,  and  the  first  terra  is  2,  and  the  common  ratio  or 
multiplier  is  3.  This  series  may  be  expressed  in  three 
forms,  as  follows: 

(1)  2  6  18  54  162  486 

(2)  2  2X3  2X3X3  2X3X3X3  2X3X3X3X3  2X3X3X3X3X3 

(3)  2  2X3  2X32  2X33  2X34  2X35 

A  comparison  of  the  corresponding  terms  of  the  three 
forms,  shows  that  each  term  of  the  series  is  composed  of 
two  factors,  viz. :  (1)  the  first  term,  and  (2)  the  common 
ratio  raised  to  a  power  whose  exponent  or  degree  is  equal  to 
the  number  of  'preceding  terms.  Hence, 

1.  The  last  term  of  a  geometrical  senes  is  equal  to  the  first 
term,  multiplied  by  the  common  ratio,  raised  to  a  power  whose 
degree  is  one  less  than  the  number  of  terms.  Conversely, 

2.  The  first  term  is  equal  to  the  last  term  divided  by  the  com¬ 
mon  ratio,  raised  to  a  power  whose  degree  is  one  less  than  the 
number  of  terms. 

3.  The  common  ratio  is  equal  to  the  root  whose  index  is  one 
less  than  the  number  of  terms,  of  the  quotient  of  the  last  term 
divided  by  the  first  term. 

454.  By  an  algebraic  process  it  may  be  shown  that 

4.  The  sum  of  an  ascending  geometrical  series  is  equal  to  the 
product  of  the  last  term  and  the  common  ratio,  less  the  first 
term,  divided  by  the  common  ratio  less  one. 

455.  When  the  number  of  terms  in  a  descending  geomet¬ 
rical  series  is  infinite,  the  last  term  is  0,  and  the  sum  of  the 
series  is  equal  to  the  first  term  divided  by  one  less  the  ratio. 


302 


COMPLETE  ARITHMETIC. 


456.  From  the  above  principles  may  be  deduced  the  fol¬ 
lowing 


FORMULAS. 


1. 

2. 

3. 

4. 

5. 


Last  term  — first  term  X  ratio  n~ 1 . 

First  term  =  last  term -—ratio n~l. 

Ratio  =  n~i/last  term  —first  term. 

C1  r  .  (last  term  X  ratio)  —  first  term 

bum  of  series  =  - - - 

ratio  —  1 

Sum  of  infinite  descending  series  =  first  term-±-(l  — ratio). 


Notes. — 1.  By  “ratio  w_1,”  in  1st  and  2d  formulas,  is  meant  the 
ratio  raised  to  a  power  whose  degree  is  the  number  of  terms  less  1. 
The  index  of  the  root,  in  the  3d  formula  (n — 1),  is  the  number  of 
terms  less  1. 

2.  In  an  ascending  series  the  ratio  is  greater  than  1,  and  in  a  de¬ 
scending  series  the  ratio  is  less  than  1. 


PROBLEMS. 

1.  What  is  the  6th  term  of  the  series  5,  10,  20?  etc. 

2.  The  first  term  of  a  geometrical  series  is  5,  the  ratio  is 
3,  and  the  number  of  terms  7 :  what  is  the  last  term  ? 

3.  The  first  term  of  a  series  is  1220,  the  ratio  -J,  and  the 
number  of  terms  6:  what  is  the  last  term? 

4.  The  last  term  of  a  series  is  64,  the  ratio  2,  and  the 
number  of  terms  10 :  what  is  the  first  term? 

5.  What  is  the  sum  of  the  series  described  in  the  4th 
problem  ?  In  the  3d  problem  ? 

6.  The  first  term  of  a  series  is  5,  and  the  sixth  term  is 
1215:  what  is  the  ratio? 

7.  The  first  term  of  a  series  is  10,  the  sixth  term  2430, 
and  the  ratio  3 :  what  is  the  sum  of  the  six  terms  ? 

8.  A  father  gave  his  son  50  cents  on  his  12th  birthday, 
and  agreed  to  double  the  amount  on  each  succeeding  birth¬ 
day  to  and  including  the  21st :  how  much  did  the  son  re¬ 
ceive  on  his  21st  birthday?  How  much  in  all? 

0.  A  man  worked  15  days  on  condition  that  he  should 
receive  1  cent  the  first  day,  5  cents  the  second  day,  and  so 


ALLIGATION. 


303 


on,  the  wages  of  each  day  being  5  times  the  wages  of  the 
previous  day :  how  much  did  he  receive  ? 

ALLIGATION. 

457.  Alligation  is  the  process  of  finding  the  average 
value  or  quality  of  a  mixture  composed  of  articles  of  dif¬ 
ferent  values  or  qualities. 

It  is  also  the  process  of  compounding  several  articles  of 
different  values  or  qualities  to  form  a  mixture  of  an  average 
value  or  quality. 

The  first  process  is  called  Alligation  Medial ,  and  the  sec¬ 
ond  Alligation  Alternate. 

Note. — The  term  Alligation  is  derived  from  the  Latin  alligare , 
to  bind  or  link.  The  term  is  applied  to  this  process  because  some  of 
the  problems  may  be  solved  by  joining  or  linking  the  numbers. 


Case  I. 

458.  Several  ingredients  of  a  mixture,  and  their  respective 
values  given,  to  find  their  average  value. 

PROBLEMS. 

1.  A  farmer  mixed  25  bushels  of  oats,  at  50  cents  a 
bushel ;  15  bushels  of  rye,  at  80  cents  a  bushel ;  and  30 
bushels  of  corn,  at  70  cents  a  bushel :  what  was  the  value 
of  a  bushel  of  the  mixture  ? 

Process. 

cts.  cts. 

50  X  25  =  1250 
80  X  15  =  1200 
70  X  30  =  2100 
70  )  4550 

65  cts.,  Ans. 

2.  A  grocer  mixed  20  pounds  of  coffee  worth  28  cents,  30 
pounds  worth  35  cents,  and  50  pounds  worth  33  cents :  what 
is  a  pound  of  the  mixture  worth  ? 


Since  the  total  value  of  the  70  bushels 
of  grain  mixed  together  was  4550  cents, 
the  value  of  1  bushel  was  of  4550  cents, 
which  is  65  cents. 


304 


COMPLETE  ARITHMETIC. 


Case  II. 

459.  The  values  of  several  articles  given,  to  find  in  what  pro¬ 
portion  they  must  he  compounded  to  make  a  mixture  of  a  given 
value. 

3.  A  grocer  has  sugars  worth  16,  18,  and  24  cents  a 
pound :  in  what  proportion  must  they  be  taken  to  make  a 
mixture  worth  20  cents  a  pound? 

I.  Solution  by  Analysis. 

On  each  pound  of  sugar  worth  16  cents  taken,  there  is  a  gain  of  4 
cents,  and  on  each  pound  at  24  cents,  there  is  a  loss  of  4  cents.  Hence, 
these  two  kinds  of  sugar  may  be  taken  in  equal  quantities,  or  1 
pound  of  each.  On  each  pound  worth  18  cents  there  is  a  gain  of  2 
cents,  and  hence  2  pounds  of  it  must  be  taken  to  offset  a  loss  of  4 
cents  on  1  pound  at  24  cents.  Hence,  the  simplest  proportionals  are 
1  lb.  at  16  cts.,  2  lb.  at  18  cts.,  and  2  lb.  at  24  cts. 


II.  Another  Solution. 


1  lb. 
1  “ 
1  “ 


at  16  cts.  selling  for  20  cts.  gains  4  cts.  1  «  ,  - 

18  “  “  20  “  “  2  cts.  /  C  ’  ga  * 

24  “  “  20  “  loses  4  cts.  .  .  4  cts.  loss. 


Taking  two  pounds  each  of  the  first  two  kinds,  the  loss  will  be  12 
cents,  and  by  taking  3  pounds  of  the  third  kind,  the  loss  will  be  12 
cents.  Hence,  the  proportionals  2,  2,  3  make  the  gains  and  losses 
equal. 


III.  Solution  by  Linking. 


20 


Note. — When  only  two  articles  of  different  values  are  given,  they 
can  be  compounded  in  but  one  way ;  but  wften  more  than  two  articles 
are  given,  they  may  be  compounded  in  an  infinite  number  of  ways. 
They  may  be  combined  two  and  two  in  such  proportions  as  to  make, 
in  each  case,  a  mixture  of  the  required  value,  and  then  these  com¬ 
pounds  may  be  united  in  any  proportions  whatever. 


APPENDIX. 


305 


4.  A  merchant  has  teas  worth  $1.25,  $1.40,  $1.60,  and 
$1.75:  how  much  of  each  kind  must  be  taken  to  make  a 
mixture  worth  $1.50? 


Case  III. 

460.  The  values  of  the  several  ingredients  of  a  mixture ,  their 
average  value ,  and  the  quantity  of  one  or  more  of  the  ingredients 
given,  to  find  the  respective  quantities  of  the  other  ingredients. 

5.  A  grocer  wishes  to  mix  100  pounds  of  coffee  at  25  cts. 
with  coffees  at  22,  28,  and  30  cts.,  making  a  mixture  worth 
27  cts.  :  how  much  of  each  kind  must  he  take  ? 

Suggestion. — Find  the  proportionals  of  the  ingredients  by  Case  II, 
and  then  multiply  each  proportional  by  the  quotient  of  100  lbs.  di¬ 
vided  by  the  proportional  for  the  coffee  worth  25  cts. 

6.  A  farmer  wishes  to  mix  60  bushels  of  corn  at  60  cts., 
with  rye  at  75  cts.,  barley  at  50  cts.,  and  oats  at  40  cts., 
to  make  a  mixture  worth  65  cts.  :  how  many  bushels  each 
of  rye,  barley,  and  oats  must  he  take  ? 

Case  IV. 

461.  The  values  of  the  ingredients,  and  the  quantity  and 
value  of  the  mixture  given,  to  find  the  quantity  of  each  ingre¬ 
dient. 

7.  How  much  gold  16  carats  fine,  18  carats  fine,  and  22 
carats  fine,  must  be  taken  to  make  12  rings  20  carats  fine, 
and  weighing  4J  pwt.  each  ? 

Suggestion. — Find  the  proportionals  by  Case  II,  and  then  divide 
the  whole  quantity  into  parts  proportional  to  these  proportionals. 

8.  How  much  sugar  worth  15  cts.,  17  cts.,  and  20  cts. 

must  be  taken  to  make  a  mixture  of  200  pounds,  worth 
18  cts.?  • 

9.  How  much  water  must  be  mixed  with  vinegar,  worth 
60  cts.  a  gallon,  to  make  90  gallons,  worth  50  cts.  a 

gallon  ? 

C.Ar.— 2 <5. 


306 


COMPLETE  ARITHMETIC. 


DUODECIMALS. 

462.  A  Duodecimal  is  a  denominate  number  in 
which  twelve  units  of  any  denomination  make  a  unit  of 
the  next  higher  denomination. 

A  duodecimal  may  be  regarded  as  a  fraction  whose  denominator 
is  a  power  of  12 ;  or  a  number  whose  scale  is  12.  The  term  is  de¬ 
rived  from  the  Latin  duodecim,  twelve. 

463.  Duodecimals  are  used  by  artificers  in  measuring 
surfaces  and  solids. 

The  foot  is  divided  into  primes ,  marked  ' ;  the  primes  into 
seconds  (")  ;  the  seconds  into  thirds  ('"),  etc.,  as  is  shown  in 
the  following 

Table. 


12  fourths  (////) 

are 

1"' 

12  thirds 

u 

1" 

12  seconds 

n 

V 

12  primes 

<< 

1  ft. 

The  accents  used  to  mark  the  different  denominations,  are 
called  Indices. 

464.  The  prime  denotes  the  twelfth  of  a  foot ;  the  second, 
the  twelfth  of  the  twelfth  of  afoot,  etc. 

When  a  duodecimal  denotes  the  area  of  a  surface,  the 
foot  is  a  square  foot;  the  prime,  the  twelfth  of  a  square  foot; 
the  second,  the  twelfth  of  a  twelfth  of  a  square  foot ,  etc. 

When  a  duodecimal  denotes  the  contents  of  a  solid,  the 
foot  is  a  cubic  foot ;  the  prime,  the  twelfth  of  a  cubic  foot,  etc. 

465.  ADDITION  AND  SUBTRACTION. 

PROBLEMS. 

1.  Add  12  ft.  8'  11",  16  ft.  10'  9",  and  24  ft.  6". 

f  12  ft.  8/  11" 

Process:  ^  16  ft.  10/  9" 

( 24  ft.  O'  6" 

53  ft.  8/  2"  Ans. 


DUODECIMALS. 


307 


2.  Add  12  ft.  9'  11"  4"',  23  ft.  7"  10"',  and  10'  6"  9'". 

3.  From  21  ft.  7'  10"  take  15  ft.  9'  4". 

t>^  /  21  ft.  r  io" 

Process:  1 15  ft>  9/  4// 

5  ft.  10/  6"  Arcs. 

4.  From  the  sum  of  30  ft.  8"  4'"  and  14  ft.  7'  10'",  take 
their  difference. 

466.  MULTIPLICATION  OF  DUODECIMALS. 

5.  Multiply  13  ft.  7'  8"  long  and  6  ft.  5'  wide. 

Multiply  first  by  5/  and  then  by 
6  ft.,  and  add  the  partial  products. 

Since  lXiV—  tV»  f?  X 
tUXtV=T72T>  etc.,  feet  x  primes 
(or  twelfths)  must  produce  ‘primes ; 
primes  by  primes,  seconds;  seconds 
by  primes,  thirds;  and,  generally, 
the  denomination  of  the  product  of  any  two  denominations  is  de¬ 
noted  by  the  sum  of  their  indices. 

6.  What  are  the  superficial  contents  of  a  board  9  ft.  7' 
4"  long  and  10'  6"  wide  ? 

7.  What  are  the  solid  contents  of  a  block  of  marble 
7  ft.  6'  long,  2  ft.  8'  wide,  and  1  ft.  4'  thick? 

Note. — The  answers  to  the  5th  and  6th  problems  are  in  square  feet 
and  duodecimal  parts  of  a  square  foot,  and  the  answer  to  the  7th 
problem  is  in  cubic  feet  and  duodecimal  parts  of  a  cubic  foot  (Art.  464). 


Process. 

13  ft.  V  8" 

6  ft.  5' _ 

5  ft.  8'  2"  4/// 

81  ft.  W  0" 

87  ft.  6'  2"  47//,  Ans. 


467.  DIVISION  OF  DUODECIMALS. 

8.  Divide  87  ft.  6'  2"  4'"  by  13  ft.  7'  8". 


Process. 

Dividend.  Divisor. 

87  ft.  6'  2"  4/7/  )  13  ft.  7/  8" 

81  ft.  10/  6  ft.  5',  (fnt. 


The  process  is  the  reverse  of 
that  in  multiplication.  For  con¬ 
venience  in  multiplying,  place  the 
divisor  at  the  right  of  the  divi¬ 
dend,  and  the  terms  of  the  quo¬ 
tient  below  those  of  the  divisor. 


5  ft.  8'  2"  4/7/ 
5  ft.  8'  2"  4/7/ 


308 


COMPLETE  ARITHMETIC. 


9.  Divide  62  ft.  11"  3'"  by  8  ft.  6'  9". 

10.  Multiply  10  ft.  5'  8"  by  3  ft.  10',  and  divide  the 
product  by  5  ft.  2'  10". 


PERMUTATIONS. 


468.  Permutations  are  the  changes  of  order,  which  a 
number  of  objects  may  undergo,  and  each  object  enter 
once  and  but  once  in  each  result. 


469.  The  diagram  at  the  right  shows 
the  number  of  permutations  of  1,  2,  and 
3  letters. 


The  letter  a  permits  no  change  of  order.  The  letter  b 
may  be  placed  before  and  after  the  letter  a ,  giving  two  (1X2) 
permutations  of  two  letters— ba,  ab.  The  letter  c  may  be 
placed  before ,  between ,  and  after  the  two  letters  ab;  and  the 
same  for  b  a,  giving  six  (1X2X3)  permutations  of  three 
letters. 

A  fourth  letter,  as  d,  may  evidently  occupy  four  different 
positions  in  each  of  the  six  combinations  of  these  letters, 
giving  twenty-four  (1X2X3X4)  permutations  of  four  letters. 

In  like  manner  it  may  be  shown  that  the  number  of  permu¬ 
tations  of  any  number  of  objects  is  equal  to  the  continued  product 
of  all  the  integers  from  1  to  the  given  number  of  objects  inclusive. 


PROBLEMS. 


1.  In  how  many  different  orders  may  6  boys  sit  on  a 
bench  ? 

2.  In  how  many  different  orders  may  all  the  letters  in 
the  word  permutation  be  written  ? 

3.  How  many  permutations  may  be  made  of  the  nine 
digits  ? 

4.  How  many  different  combinations  of  eight  notes  each 
may  be  made  of  the  octave? 


RULES  OF  MENSURATION. 


309 


I 


ANNUITIES. 

470.  An  Annuity  is  a  sum  of  money,  payable  annu¬ 
ally,  for  a  given  number  of  years,  for  life,  or  forever. 
The  term  is  also  applied  to  sums  of  money  payable  at  any 
regular  intervals  of  time. 

471.  A  Certain  Annuity  is  an  annuity  that  is  payable  for 
a  given  number  of  years. 

A  Contingent  Annuity  is  an  annuity  payable  for  an  uncer¬ 
tain  period,  as  during  the  life  of  a  person. 

A  Perpetual  Annuity  is  one  that  continues  forever. 

472.  An  Immediate  Annuity  is  an  annuity  whose  payment 
begins  at  once. 

A  Deferred  Annuity  is  an  annuity  whose  payment  begins 
at  a  future  time. 

473.  The  Forborne  or  Final  Value  of  an  annuity  is  the 
sum  of  the  compound  amounts  of  all  its  payments,  from  the 
time  each  is  due  to  the  end  of  the  annuity. 

The  Present  Value  of  an  annuity  is  the  present  worth  of 
the  forborne  or  final  value. 

Note. — The  principal  applications  of  the  subject  of  annuities  are 
in  leases,  life  estates,  rents,  dowers,  life  insurance,  etc. ;  and  the  prob¬ 
lems  arising  are  readily  solved  by  means  of  tables  which  give  the 
present  and  final  values  of  $1  at  the  usual  rates  of  interest.  A  full 
discussion  of  the  principles  involved  in  the  construction  of  these  tables, 
can  not  well  be  presented  in  a  school  arithmetic. 


RULES  OF  MENSURATION. 

474.  Surfaces  and  Lines. 

1.  To  find  the  area  of  a  rectangle,  Multiply  the  length  by 
the  width. 

2.  To  find  either  side  of  a  rectangle,  Divide  the  area  by 
the  other  side. 


310 


COMPLETE  ARITHMETIC. 


3.  To  find  the  area  of  a  triangle,  Multiply  the  base  by  one 
half  of  the  altitude. 

4.  To  find  the  area  of  any  quadrilateral  having  two  sides 
parallel,  Multiply  one  half  of  the  sum  of  the  two  parallel  sides 
by  the  perpendicular  distance  between  them. 

5.  To  find  the  circumference  of  a  circle, 

1.  Multiply  the  diameter  by  3.1416.  Or, 

2.  Divide  the  area  by  one  fourth  of  the  diameter. 

6.  To  find  the  area  of  a  circle, 

1.  Multiply  the  square  of  the  diameter  by  .7854.  Or, 

2.  Multiply  the  square  of  the  radius  by  3.1416.  Or, 

3.  Multiply  the  circumference  by  one  half  of  the  radius. 

7.  To  find  the  diameter  of  a  circle,  whose  area  is  given, 
Divide  the  area  by  .7854,  and  extract  the  square  root  of  the  quo¬ 
tient. 

8.  To  find  the  side  of  the  largest  square  that  can  be  in¬ 
scribed  in  a  circle,  Multiply  the  radius  by  the  square  root  of  2. 

9.  To  find  the  side  of  the  largest  equilateral  triangle 
that  can  be  inscribed  in  a  circle,  Multiply  the  radius  by  the 
square  root  of  3. 

10.  To  find  the  area  of  an  ellipse,  the  two  diameters  be¬ 
ing  given,  Multiply  the  product  of  the  two  diameters  by  .7854. 

11.  To  find  the  surface  of  a  sphere, 

1.  Multiply  the  circumference  by  the  diameter.  Or, 

2.  Multiply  the  square  of  the  diameter  by  3.1416. 

12.  To  find  the  entire  surface  of  a  right  prism  or  right 
cylinder,  Multiply  the  perimeter  or  circumference  of  the  base  by 
the  height ,  and,  to  the  product,  add  the  surface  of  the  two  bases. 

13.  To  find  the  convex  surface  of  a  pyramid  or  cone, 
Multiply  the  perimeter  or  circumference  of  the  base  by  one  half 
the  slant  height. 

14.  To  find  the  hypotenuse  of  a  right-angled  triangle, 


RULES  OF  MENSURATION. 


311 


Extract  the  square  root  of  the  sum  of  the  squares  of  the  other 
two  sides. 

15.  To  find  the  base  or  the  perpendicular  of  a  right- 
angled  triangle,  Extract  the  square  root  of  the  difference  be¬ 
tween  the  square  of  the  hypotenuse  and  the  square  of  the  other 
side. 


475.  Contents  of  Solids. 

1.  To  find  the  solid  contents  of  a  rectangular  solid,  Mul¬ 
tiply  the  length ,  width,  and  thickness  together. 

2.  To  find  either  dimension  of  a  rectangular  solid,  Divide 
the  solid  contents  by  the  product  of  the  other  two  dimensions. 

3.  To  find  the  solid  contents  of  a  cylinder,  Multiply  the 
area  of  the  base  by  the  altitude. 

4.  To  find  the  solid  contents  of  a  sphere, 

1.  Multiply  the  cube  of  the  diameter  by  .5236.  Or, 

2.  Multiply  the  surface  by  one  third  of  the  radius. 

5.  To  find  the  solid  contents  of  a  cone  or  pyramid, 
Multiply  the  area  of  the  base  by  one  third  of  the  altitude. 

6.  To  find  the  solid  contents  of  the  frustum  of  a  cone 
or  pyramid,  To  the  sum  of  the  areas  of  the  two  bases,  add  the 
square  root  of  their  product,  and  multiply  the  result  by  one  third 
of  the  altitude. 


Board  Measure. 

1.  To  measure  lumber  1  inch  or  less  in  thickness,  Multiply 
the  length  in  feet  by  the  width  in  inches,  and  divide  the  product 

by  12. 

2.  To  measure  lumber  more  than  1  inch  in  thickness,  as 
planks,  joists,  etc. ,  Multiply  the  number  of  square  feet  in  one 
surface  by  the  thickness  in  iiiches. 

Note. — Boards  1  inch  or  less  in  thickness  are  sold  by  the  square 
foot,  surface  measure;  but  lumber  more  than  1  inch  in  thickness  is 
measured  by  finding  the  number  of  square  feet  in  one  surface,  and 
multiplying  the  result  by  the  thickness  in  inches. 


312 


COMPLETE  ARITHMETIC. 


FOREIGN  EXCHANGE. 

476.  A  Foreign  Bill  of  Exchange  is  a  draft 
drawn  in  one  country  and  payable  in  another.  (Art.  328). 

Foreign  Bills  are  expressed  in  the  currency  of  the  country  on 
which  they  are  drawn.  They  are  issued  in  sets  of  three,  of  the  same 
tenor  and  date,  called  the  First,  Second,  and  Third  of  Exchange, 
and  are  sent  by  different  mails  to  avoid  delay  in  case  of  miscarriage. 
When  one  is  paid,  the  others  are  void. 

477.  The  Far  of  Exchange  is  the  comparative 
value  of  the  currencies  of  two  countries.  (Art.  329). 

The  commercial  value  of  foreign  exchange  may  be  higher,  equal 
to,  or  lower  than  the  par  of  exchange.  A  bill  payable  in  sixty 
days  costs  less  than  a  bill  payable  on  sight,  or  in  three  days,  called 
“short  sight.” 

The  commercial  or  quoted  value  of  exchange  is  used  in  finding 
the  cost  of  a  foreign  bill. 


EXCHANGE  ON  ENGLAND. 

478.  Bills  between  the  United  States  and  England  are 
expressed  in  sterling  money,  and  are  drawn  on  London. 
They  are  called  Sterling  Bills. 

The  legal  or  par  value  of  a  pound  sterling  is  $4.8665,  and  this  is 
now  custom-house  value.  The  commercial  or  exchange  value  is 
now  quoted  in  dollars  and  cents,  gold.  The  gold  coin,  whose  value 
is  £1,  is  called  a  Sovereign. 

PROBLEMS. 

1.  What  will  a  bill  on  London  for  £448  11  s.  cost  in 
New  York,  when  sterling  exchange  is  quoted  at  4.86§? 

Process. 

£448  11s.  =  £448.55. 

$4.86$  X  448.55  =  $2182.943,  cost  of  bill. 

Since  £1  is  worth  $4.86$,  £448.55  are  worth  $4.86$  X  448.55, 
which  is  $2182.943. 


FOREIGN  EXCHANGE. 


313 


2.  What  will  a  sterling  bill  for  £219  10s.  6d.  cost  in 
New  York,  when  sterling  exchange  is  quoted  at  4.91J? 

3.  What  will  a  bill  on  London  for  £200  12s.,  payable  in 
60  days,  cost  in  New  York,  when  sterling  exchange  is 
quoted  at  4.85f  ? 

Note. — In  all  exchange  problems  in  this  edition,  the  gold  quo¬ 
tations  are  omitted. 

4.  What  will  a  sight  draft  on  London  for  £300  8s.  cost 
in  New  York,  when  sterling  exchange  is  quoted  at  4.88? 

5.  What  will  a  sight  draft  on  London  for  £250  cost  a 
merchant  in  Cincinnati,  when  sterling  exchange  is  quoted  at 
4.86,  and  the  broker’s  commission  is  of  cost  of  draft  in 
New  York? 

6.  What  will  be  the  cost  of  the  following  bill  when  ster¬ 
ling  exchange  is  quoted  at  4.85J? 

£1000.  New  York,  Jan.  10,  1876. 

Sixty  days  after  sight  of  this  First  of  Exchange  (Second 
and  Third  of  same  tenor  and  date  unpaid)  pay  to  the  order 
of  Wilson,  Hinkle  &  Co.  One  Thousand  Pounds,  value  re¬ 
ceived,  and  charge  to  account  of 

August  Belmont  &  Co. 

To  Brown ,  Shipley  &  Co.,  ) 

London.  j 

7.  What  amount  of  sterling  exchange  can  be  bought  for 
$1080.45  in  gold,  when  sterling  exchange  is  quoted  at  4.90? 

Process. 

$1080.45  — t-  $4.90  =  220.5;  £220.5  =  £220  10s. 

8.  How  large  a  draft  on  London  can  be  bought  in  Chicago 
for  $2195.475,  when  sterling  exchange  is  quoted  at  4.86§,  and 
the  broker’s  commission  is  \^/0  of  cost  of  draft  in  New  York? 

9.  How  large  a  sight  draft  on  London  can  be  bought  in 
New  York  for  $1174.20,  when  sterling  exchange  is  quoted 
at  4.89J? 

C.  Ar.— 27. 


314 


COMPLETE  ARITHMETIC. 


479.  Rules. —  1.  To  find  the  cost  of  sterling  exchange, 
Multiply  the  cost  of  £1  by  the  number  of  pounds  denoting  the  face 
of  the  bill. 

Note. — When  the  face  of  the  bill  contains  shillings  and  pence, 
reduce  them  to  the  decimal  of  a  pound.  (Art.  171.) 

2.  To  find  the  amount  of  sterling  exchange  that  can  be 
bought  for  a  given  sum  of  United  States  money,  Divide  the 
given  sum  of  money  by  the  cost  of  £1  of  exchange. 


EXCHANGE  ON  FRANCE. 

480.  The  New  York  quotations  of  exchange  on  Paris  give 
the  number  of  francs  and  centimes  which  are  equal  in  ex¬ 
change  to  $1  of  United  States  money  (gold).  The  centimes 
are  usually  expressed  as  hundredths. 

Quotations  on  Antwerp  and  Switzerland  are  also  in  francs.  Quota¬ 
tions  on  Amsterdam  are  in  guilders,  worth  about  41  cents. 

The  value  of  a  franc  (Louis  Napoleon)  is  $.192  nearly,  $1  being 
equal  to  5  francs  and  14|  centimes.  The  custom-house  value 
is  $.193. 


PROBLEMS. 

1.  What  will  be  the  cost  of  a  bill  on  Paris  for  3870  fr., 
when  Paris  exchange  is  quoted  in  New  York  at  5.16? 

Process. 

3870  fr.  -r-  5.16  fr.  =  750 ;  $1  X  750  =  $750. 

Since  $1  will  buy  5.16  fr.,  it  will  take  as  many  times  $1  to  buy 
3870  fr.  as  5.16  fr.  is  contained  times  in  3870  fr.,  which  is  750. 

2.  What  will  a  draft  on  Paris  for  6475  fr.  cost  in  New 
York,  when  Paris  exchange  is  quoted  at  5.18? 

3.  What  will  a  bill  on  Paris  for  5330  fr.  cost  in  New 
York  when  Paris  exchange  is  quoted  at  5.12J? 


FOREIGN  EXCHANGE. 


315 


4.  What  amount  of  exchange  on  Paris  can  be  bought 
for  $1500,  when  Paris  exchange  is  quoted  at  5. 14J? 

Process. 

5.14|  fr.  X  1500  =  7717.5  fr.,  Ans. 

5.  How  large  a  draft  on  Paris  can  be  bought  in  New 
York  for  $2432,  when  Paris  exchange  is  quoted  at  5.16§? 

6.  What  will  be  the  cost  of  a  bill  on  Antwerp  for 
6418f  fr.,  when  exchange  is  quoted  at  5.13J? 

7.  What  amount  of  exchange  on  Switzerland  can  be 
bought  for  $650  when  exchange  is  quoted  at  5.15? 

481.  Rules. — 1.  To  find  the  cost  of  exchange  on  Paris, 
Divide  the  number  of  francs  in  the  face  of  the  bill  by  the  number 
of  francs  that  equal  $1  of  exchange . 

2.  To  find  the  amount  of  exchange  on  Paris  that  can  be 
bought  for  a  given  sum  of  money,  Multiply  the  number  of 
francs  that  equal  $1  of  exchange  by  the  number  denoting  the 
given  sum  of  money. 


EXCHANGE  ON  GERMANY. 

482.  Exchange  on  Germany  is  quoted  at  so  many  cents 
per  four  reichsmarks  (marks). 

The  par  value  of  a  reichsmark,  the  new  German  coin,  is  $.243, 
or  about  11  francs. 


PROBLEMS. 

1.  What  will  be  the  cost  of  a  sight  bill  on  Hamburg  for 
2240  marks,  when  exchange  is  quoted  in  New  York  at  .96J? 


Process. 

$.96}  X  2240  -s-  4  ==  $539,  cost. 


316 


COMPLETE  ARITHMETIC. 


2.  What  would  be  the  cost  of  a  sight  bill  on  Berlin  for 
1680  marks,  when  exchange  is  quoted  at  .96J? 

3.  What  would  be  the  cost  in  St.  Louis  of  a  sight  draft 
on  Hamburg  for  3200  marks,  when  exchange  is  quoted  at 
.96J,  and  the  broker’s  commission  is  \^0  ? 

4.  What  amount  of  exchange  on  Frankfort  can  be  bought 
for  $1752.60,  when  exchange  is  quoted  at  .95J? 

Process. 

$1752.60  -r-  $.95J  X  4  =  7360,  marks. 

5.  What  amount  of  exchange  on  Berlin  can  be  bought 
for  $324,  when  exchange  is  quoted  at  .94J? 

6.  What  will  a  draft  for  960  marks  cost,  when  exchange 
is  quoted  at  .95|? 

7.  How  large  a  draft  on  Frankfort  can  be  bought  for 
$514.35,  when  exchange  is  quoted  at  .95J? 

8.  What  amount  of  exchange  on  Hamburg  can  be  bought 
for  $231,  when  exchange  is  quoted  at  .96,  and  the  broker’s 
commission  is  \%  ? 

483.  Rules. — 1.  To  find  the  cost  of  exchange  on  Ger- 
many,  Multiply  the  cost  of  four  marks  by  the  number  of  marks 
in  the  face  of  the  bill ,  and  divide  the  result  by  4. 

2.  To  find  the  amount  of  exchange  on  Germany  that  can 
be  bought  for  a  given  sum  of  money,  Divide  the  given  sum 
of  money  by  the  cost  of  four  marks ,  and  multiply  the  result  by 
4.  Or,  Divide  the  given  sum  of  money  by  the  cost  of  1  mark. 


.ANSWERS 


TO 

THE  WRITTEN  PROBLEMS. 


N.  B. — The  last  answer  is  given  when  a  problem  has  several  an¬ 
swers,  and  also  when  several  problems  are  united. 


NOTATION. 

Page  11. 


9. 

40,605. 

15. 

4,014,045,000. 

10. 

700,007. 

16. 

65,000,006,050. 

11. 

5,005,500. 

17. 

850,049,000,000. 

12. 

60,060,060. 

18. 

17,070,000,700,400. 

13. 

700,700,700. 

19. 

56,000,016,000,090. 

14. 

560,068,000. 

20. 

7,000,085,000,000,204. 

ADDITION. 


Page  14. 

19.  8,984,342. 

26.  598. 

13. 

108,657. 

20.  3,578,392  sq.  m. 

27.  732. 

14. 

442,555. 

21.  258. 

28.  803. 

15. 

63,077,833. 

Page  16. 

29.  631. 

16. 

74,467,648. 

22.  383. 

30.  865. 

17. 

12,369  bush. 

23.  512. 

31.  636. 

Page  15. 

24.  462. 

32.  633. 

18. 

443,275  sq.  m. 

25.  649. 

SUBTRACTION. 

Page  18. 

20.  1,116,942  sq.  m. 

Page  19. 

16. 

$4,075. 

21.  914,054  sq.  m. 

26.  $800. 

17. 

49,894,136  m. 

22.  56,077,528  bush. 

27.  6,890  bush. 

18. 

429,559. 

23.  $467. 

28.  All,  1802  A. 

19. 

35,965. 

24.  $2,330. 

29.  26,956. 

25.  $1,032. 

317 


21-36 


COMPLETE  ARITHMETIC. 


Page  21. 

11.  2,499,120. 

12.  230,668,800. 

13.  503,232,000. 

14.  364,800,000,000. 

15.  17,424,000  ft. 

16.  1,572,480  m. 

17.  25,600,000  A. 

18.  28,500. 

19.  Gained  $798. 

Page  23. 

7.  4,560,000. 


Page  26. 

15.  54. 

16.  233,  with  20  R. 

17.  7,  with  600  R. 

18.  1,  with  109,304  R. 

19.  3,464. 

20.  8,743. 

21.  4,567. 

22.  41  cars. 

23.  5m.  2,700  ft. 

Page  27. 

24.  205  h. 


MULTIPLICATION. 


8.  305,000,000. 

9.  347,000,000,000. 

10.  88,900,000. 

16.  15,300. 

17.  16,200. 

18.  84,400. 

Page  24. 

19.  13,549,333£. 

20.  867,000. 

21.  1,362,000. 

22.  691,066|. 

DIVISION. 

25.  548,  with  128  R. 

Page  29. 

8.  356. 

9.  46,  with  35  R. 

10.  38,  with  4,602  R. 

11.  95. 

Page  30. 

1 8.  9,  with  200  R. 

1 9.  24,  with  800  R. 

20.  9,  with  10,800  R. 

2 1.  2,  with  1,600  R. 
28.  125. 


23.  45,766. 

24.  5,666,328. 

25.  40,740,411. 

26.  86,772,642. 

27.  61,870,306,330. 

28.  322,096. 

Page  25. 

29.  652,919. 

30.  94,240. 

31.  25,629,438. 

32.  4,432,246. 

33.  613,566. 


Page  31. 

29.  26,  with  190  R. 

30.  45,  with  300  R. 

32.  761,  with  39  R. 

33.  190,  with  28  R. 

34.  387,  with  13  R. 

35.  57,  with  39  R. 

Page  32. 

36.  480  with  7  R- 

37.  1,487,  with  26  R. 

38.  5,203  with  40  R. 

39.  3,604,  with  16  R. 

40.  433,  with  3  R. 


PROPERTIES  OF  NUMBERS. 


Page  33. 

15.  2,  2,  2,  2,  2,  5. 

1 6.  5,  5,  7. 

17.  2, 2, 2,  2,  2, 2, 2, 2. 

18.  5,  5,  13. 

19.  2,  3,  5,  11. 

20.  2,  2,  3,  5,  7. 

21.  2,  3,  7,  11. 

22.  2,  5,  7,  7. 

23.  2,  3,  3,  3,  11. 


24.  2,  2,  11,  17. 

25.  3,  3,  7,  11. 

26.  3,  5,  7,  11. 

Page  34. 

27.  3,  3. 

28.  5, 2. 

29.  2,  2,  2,  2,  2. 

30.  2,  5,  5. 

31.  5,5. 

32.  2,  2,  3,  3. 


Page  35. 

34.  7. 

35.  12, 

36.  55. 

Page  36. 

38.  b 

39.  8. 

40. 

4 1.  54  cts. 

42.  9  men. 


318 


ANSWERS. 


38-53 


GREATEST  COMMON  DIVISOR. 


Page  38. 

20.  48. 

29. 

39  lbs. 

12. 

12. 

21.  37. 

30. 

25. 

13. 

63. 

22.  l. 

31. 

120. 

14. 

48. 

23.  252. 

32. 

72. 

15. 

28. 

24.  14. 

33. 

7. 

16. 

42. 

25.  192. 

34. 

8. 

17. 

128. 

26.  57. 

35. 

5. 

18. 

48. 

27.  $52. 

36. 

13. 

19. 

4. 

28.  $165. 

37. 

1. 

LEAST  COMMON  MULTIPLE. 

Page  41. 

18.  300. 

25. 

756. 

12. 

120. 

19.  $720. 

26. 

720. 

13. 

126. 

20.  420. 

27. 

1,200. 

14. 

480. 

21.  180. 

28. 

1,890. 

15. 

108. 

22.  280. 

29. 

3,360. 

16. 

144. 

23.  180. 

30. 

2,520. 

17. 

210. 

24.  600. 

FRACTIONS. 

Page  46. 

32.  191. 

60. 

5 

13* 

8. 

2923 

~2~' 

33.  9. 

61. 

3 

2* 

9. 

988 

'T2~* 

34.  I7f. 

62. 

5 

7* 

10. 

4061 

22  * 

35.  38y. 

63. 

1  3 

'7'* 

11. 

13  3 

T2~- 

36.  46^. 

64. 

5 

4* 

12. 

112  1 
"T2-' 

37.  109A. 

Page  49„ 

13. 

618  7 

3  0  * 

38.  12gf3. 

65. 

1  8 

TT- 

Page  47. 

39.  53jtj. 

Page  50. 

14. 

5803 

4  0  * 

40.  2,016f. 

87. 

32  66  34  69 

72)  72)  72)  72* 

15. 

343  3 
~22~' 

Page  48. 

88. 

80  72  100  105 

120)  120)  120)  120' 

16. 

20409 

51.  f 

89. 

42  3  2  33.  2  6 

60)  60)  60)  60* 

17. 

2  4  6  1  3 

4T  * 

52.  If. 

90. 

49  32  33  34 

84)  24)  24)  84- 

18. 

HP- 

53.  |~f* 

Q1  24  55  33  58  31 

v  A.  62)  22)  62)  20)  60* 

19. 

54. 

92. 

4025  2898  3542 

7242)  7242)  7242) 

20. 

HP- 

55.  f. 

1380  6363 

7242)  7242* 

28. 

16tV 

56.  fl* 

Page  52. 

29. 

27. 

57.  1- 

107. 

2 

2' 

30. 

17f. 

58.  2§* 

108. 

7- 

31.  39*. 

59. 

109. 

4' 

319 


COMPLETE  ARITHMETIC. 


52-59 


110. 

8 

Z‘ 

30. 

4** 

42. 

7f 

111. 

20 

21- 

31. 

193*. 

43. 

1  7 

6  0* 

112. 

4 

z- 

32. 

288f. 

44. 

2 

Z‘ 

113. 

3 

2* 

33. 

274**. 

Page  57. 

114. 

49 

¥T)" 

34. 

174. 

45. 

8  1 

63" 

115. 

128 

8  • 

35. 

65if. 

46. 

3 

20" 

116. 

5 

3* 

36. 

$20.37*. 

Page  58. 

117. 

-tit  gal. 

37. 

68**. 

11. 

4*. 

118. 

12§. 

38. 

Q 1  7 
dZ0' 

12. 

5** 

119. 

$63XV 

Page  55. 

13. 

5f. 

120. 

12  21  22 

3  0)  3  0)  3  0 ' 

13. 

3 

TT" 

14. 

7f* 

121. 

150  225  400 

"12)  12)12" 

14. 

43 

90" 

15. 

9  1 
■"1  2* 

122. 

3  6 

65" 

Page  56. 

16. 

91  5 
■"16" 

123. 

15  20  14 

2¥)  2T)  2T" 

15. 

17. 

5*. 

124. 

5  33  2 

¥)  '6  )  6" 

16. 

5 

3  6" 

18. 

4**. 

125. 

28  36  336 

6  3)  6  3)  6  3  " 

17. 

1  7 

ZZ’ 

19. 

2,250. 

126. 

15  30  25 

12)  12)  12" 

18. 

1 1 

TOO" 

20. 

3,648. 

127. 

14  48  135 

24)  2?)  2?" 

19. 

5 

¥2) 

21. 

$166. 

128. 

10  5  450  1000 

120)  120)  1  2  0  ) 

20. 

19 

60" 

22. 

$3,466*. 

2  7  6 

120" 

21. 

1  3 

13  2" 

Page  59. 

Page  53. 

22. 

3  5 

1  OZ’ 

34. 

514. 

12. 

2f. 

23. 

_7_ 

3^" 

35. 

50*. 

13. 

014 

^ZZ- 

24. 

_7_ 

3  0" 

36. 

29*. 

14. 

2tVo" 

25. 

7 

3¥" 

37. 

254*. 

15. 

Iff* 

26. 

93  9 

Z¥0" 

38. 

623*. 

Page  54. 

27. 

5 

Y2" 

39. 

465*. 

16. 

iH- 

28. 

1  2 

TT* 

40. 

432*. 

17. 

iff- 

29. 

92**. 

41. 

1,702*. 

18. 

Ol  7 

■“3  6 * 

30. 

46*. 

42. 

522f. 

19. 

2*. 

31. 

19*. 

43. 

13,736*. 

20. 

2-b 

32. 

17H- 

44. 

572. 

21. 

2f. 

33. 

88**. 

45. 

693. 

22. 

031 

AZZ' 

34. 

$.70f. 

46. 

808. 

23. 

O  91 

ZT62" 

35. 

$.63f. 

47. 

8,223f. 

24. 

9  5 
^36" 

36. 

1  3 

2?" 

48. 

8,649. 

25. 

*• 

37. 

9**. 

49. 

13,533*. 

26. 

If*" 

38. 

i*. 

50. 

45,196*. 

27. 

If. 

39. 

119 

120" 

51. 

17,427*. 

28. 

3f. 

40. 

ItV 

52. 

42,745*. 

29. 

lit* 

41. 

HI* 

320 


ANSWERS. 


60-78 


Page  60. 

34.  12. 

77.  9. 

62.  ff. 

35.  36. 

78.  AV 

63.  !?. 

36.  40 

79.  3§. 

64-  tV 

Page  64. 

80.  Iff* 

65.  f. 

45.  f. 

Page  71. 

66.  f. 

46.  lsV* 

51.  5. 

67.  sV 

47.  f. 

5  2.  ts* 

68.  If. 

48.  f. 

53.  411. 

69.  b 

49.  lb 

54.  81. 

70.  8 b 

50.  lyV 

55.  Iff. 

71.  24f. 

5  1.  T5’ 

56.  Iff 

72.  221 1. 

52.  5. 

57.  504f 

73.  28. 

53.  b 

58.  if 

74.  $||. 

54.  b 

59.  188 1  A. 

7  5.  34|  cts. 

55.  4f. 

60.  26  sq.  rd. 

76.  $3f. 

56.  2if 

61.  1311  h. 

77.  $253§. 

Page  65. 

62.  121 T. 

Page  61. 

5  7.  5f  months. 

63.  $3,132  b 

78.  $7,729f 

5  8-  15f  bu. 

64.  14f  yds. 

79.  Iff* 

59.  46|  yds. 

65.  105. 

80.  $2,811. 

60.  6232  h. 

66.  115|  m. 

Page  62. 

61.  25f  A. 

67.  $1,375. 

11.  ST* 

62.  44f. 

68.  $14,175. 

12.  S23* 

63.  25ff. 

Page  72. 

1 3.  t<j* 

Page  66. 

69.  390. 

1 4.  fV* 

65.  xV 

70.  $14,616. 

1 5.  s3r* 

66.  sV* 

71.  $2,555f. 

1  6.  7J- 

67.  27. 

70  49 

4  6.  160* 

17.  sV 

68.  !• 

73.  sV 

1  8.  ST* 

69.  |. 

74.  A,  220  A;  B,  176 

19.  f. 

70.  lb 

A. 

Page  63. 

71.  20. 

75.  A’s  $2,310;  B’s 

28.  36. 

72.  so* 

$2,800  ;C’s  $1,050. 

29.  45. 

73.  2f. 

76.  $35,200. 

30.  75. 

74.  sV 

77.  $4,875. 

31.  1231. 

75.  f. 

78.  8  of  each. 

32.  169t77. 

33.  297ff. 

76.  2f. 

79.  $9,000,  estate. 

DECIMAL  FRACTIONS. 

Page  78. 

59.  .040034 

61.  .0000615 

58.  .0205 

60.  .02004 

62.  600.0015 

321 


78-87 


COMPLETE  ARITHMETIC. 


63.  15.015 

64.  .00300303 

65.  .5000085 

66.  .00012 

67.  400.000465 

68.  25.025 

69.  5000.005 

70.  375.000000375 

Page  79. 

71.  .30046 

72.  .001000045 

73.  80040.0306 

74.  15000.0015 

75.  75.005043 

76.  1000000.000001 

7.  .0674000 

8.  .07500 

9.  62.700 

10.  5.3300 

11.  3.00 

12.  45.0000 

13.  .045 

14.  5.24 

Page  80. 

19.  | 

20.  f 

21.  5% 

22.  ts 

23. 

24.  59o 

25.  sis 

26.  ts 

27.  fo 

QQ  141 

<50.  50^ 

29.  3|i 

30.  37f 

31.  62?V 

32.  37f 

33.  56| 

34.  247i 

35.  16f 

36.  214^!^ 


Page  81. 

42.  .625 

43.  .5625 

44.  .04 

45.  .78125 

46.  .512 

47.  .64 

48.  1.28 

49.  3  625 

50.  .096 

51.  .075 

52.  .0875 

53.  .095 

54.  .325 

55.  .0175 

56.  .092 

57.  .0032 

58.  .0013! 

59.  .04375 

60.  12.15 

61.  25.032 

62.  37.1625 

63.  .00831 

64.  -076f 

65.  .126^ 

Page  82. 

2.  210.08595 

3.  111.0188 

4.  $267.322f 

5.  120.0905! 

6.  .2806484 

7.  .0252077 

8.  148.58!  rd. 

9.  5.00!  lb. 

10.  22.84  in. 

Page  83. 

3.  2.0425 

4.  .61625 

5.  11.9995 

6.  .594 

7.  .043956 

8.  .026095 

322 


9.  .005005193 

10.  4.95  miles. 

1 1.  2.55  in. 

12.  .22° 

Page  84. 

9.  4.875 

10.  .2795 

11.  .23328 

12.  1.152 

13.  .3136 

14.  1.1772 

15.  2.048 

16.  .5454 

17.  1.344 

18.  4. 

19.  2.55 

20.  640. 

21.  1.08 

22.  49.45 

23.  .75375 

24.  256. 

25.  .00000943 

26.  625. 

Page  85. 

27.  3406. 

28.  48. 

29.  25.6 

Page  87. 
12.  18. 

13.  4. 

14.  8. 

15.  30. 

16.  2500. 

17.  20. 

18.  150. 

19.  24. 

20.  .05 

21.  .08 

22.  2.07 

23.  .27 

24.  .0066 
25.  790. 


1 


ANSWERS. 


87-104 


26. 

900. 

37.  192. 

4. 

9 

27(70 

27. 

.009 

38.  1500. 

5. 

1.08 

28. 

.009 

39.  2294.11^ 

6. 

63.5475 

29. 

3413J. 

40.  .00025 

7. 

39.056875 

30. 

.001024 

42.  .48375 

8. 

320. 

31. 

.00005 

43.  .00545 

9. 

399.514 

32. 

20000. 

44.  .00005 

10.  $177.66| 

33. 

.00001 

Page  88. 

11. 

.264,  or  1031.25. 

34. 

100000. 

1.  .024 

12. 

105. 

35. 

.00005 

2.  .0028 

13. 

.000064 

36. 

36. 

q  13 

O.  ;ny 

14. 

50000000000. 

UNITED  STATES  MONEY. 

Page  90. 

10.  $3361.25 

Page  96. 

5. 

$10.50 

11.  $1053.10 

9. 

$565.50 

6. 

$40,605 

12.  $225. 

10. 

$1000. 

7. 

$100,374 

13.  $378.60 

11. 

$4004. 

8. 

$25,005 

Page  92. 

12. 

$2343.75 

9. 

$.065 

14.  $83.50 

13. 

$480. 

10. 

$.104 

8.  $1010.25 

14. 

242  dozens. 

11. 

35000  cts. 

9.  $9.22+ 

15. 

360  yards. 

12. 

165000  m. 

10.  $8666f 

16. 

$1,969 

13. 

17  m. 

11.  $40.75 

17. 

$1.9625 

14. 

4008  cts. 

12.  60  carriages. 

18. 

$31,825 

15. 

10000  m. 

Page  93. 

Page  97. 

16. 

$15. 

1 3.  94  tons. 

19. 

$35.4375 

17. 

$15. 

14.  $68.55 

20. 

$2,246+ 

18. 

45  cts. 

15.  $527.05 

Page  98. 

19. 

25080  m. 

16.  $163.20 

2. 

$469,125 

20. 

100010  m. 

Page  94. 

3. 

$40,946 

Page  91. 

1.  $236.35 

4. 

$160,758 

7. 

$271.64 

2.  $389.19 

Page  99. 

8. 

$37,775 

3.  $4569.02 

5. 

$22,017 

9. 

$617.20 

4.  $53463.64 

6. 

$39.68£ 

MENSURATION. 

Page  103. 

1 4.  7035  sq.  ch. 

18. 

165  yards. 

11. 

862£  sq.  ft. 

15.  13000  sq.ft. 

19. 

904.7  sq.  yd. 

12. 

404.625  sq.  yd. 

16.  66  yards. 

20. 

15  inches. 

13. 

f  897f-  sq.  ft. 

Page  104. 

21. 

113.0976  sq.  in. 

450  sq.  ft. 

1  7.  24  rods. 

22. 

314.16  sq.  ft. 

323 

106-113 


COMPLETE  ARITHMETIC. 


Page  106. 

11. 

1953.125  cu.  ft. 

14. 

76545  bricks. 

1102£  cu.  ft. 

12. 

5040  cu.  ft. 

15. 

225  cans. 

166.375  cu.  yd. 

13. 

8  ft. 

16. 

2827.44  cu.  in. 

DENOMINATE  NUMBERS. 


Page  109. 

30. 

1.092  h. 

32. 

9911  dr. 

31. 

.04  lb. 

33. 

778  pt. 

32. 

.02  rd. 

34. 

23983  yd. 

33. 

.09625  mi. 

35. 

1902  P. 

34. 

6.144  pt. 

36. 

7  bu.  3  pk.  3qt.  1  pt. 

35. 

i  l 

37. 

10  gal.  1  pt. 

Page  112. 

38. 

1  mi.  7  fur.  22  rd.  5  yd.  2  ft. 

36. 

6  72  cir. 

39. 

3°  27'  40". 

37. 

979 J  min. 

40. 

8  h.  31  min.  24  sec. 

38. 

128  h. 

41. 

57040  d. 

39. 

TeVo  ^a* 

42. 

51320  P. 

40. 

.07  A. 

43. 

2  cwt.  55  lb.  5  oz. 

41. 

108.96  cd.  ft. 

44. 

2  mi.  3  fur.  18  rd.  1  yd. 

42. 

9.472  dr. 

45. 

867240  in. 

43. 

326160  gr. 

46. 

1  mi.  6  fur.  12  rd.  4  yd.  1  ft.  8  in. 

44. 

xi  Pt- 

47. 

26  cd.  7  cd.  ft.  10  cu.  ft. 

45. 

12830.4  ft. 

48. 

160  bu.  4  qt. 

46. 

3.24  £. 

49. 

140160  h. 

47. 

.00026|  mi. 

50. 

31622400  sec. 

48. 

7  rln 

3  6  0  Ucl- 

51. 

2167200  min. 

49. 

21.12  yd. 

52. 

236.8  pt. 

50. 

.00045T5r  mi. 

53. 

74.25  ft. 

Page  113. 

54. 

43.8  oz. 

14. 

6  fur.  8  rd.  4  yd.  2  ft.  8  in. 

55. 

56436". 

15. 

2  da.  22  h. 

56. 

163°  28'  7". 

16. 

5  oz.  12  pwt. 

57. 

64944  ft. 

17. 

4  yd.  1  ft.  4|  in. 

58. 

52  w.  1^  da. 

18. 

3  R.  26 1  P. 

59. 

4  A.  1  R.  26.35  P. 

19. 

112  cu.  ft. 

60. 

8064  A. 

20. 

13  oz.  9.6  dr. 

Page  111. 

21. 

6  cwt.  50  lb. . 

26. 

d1’- 

22. 

3  in. 

27. 

19J  in. 

23. 

3  qt.  1  pt.  2  gi. 

28. 

220  pwt. 

24. 

56  lb.  4  oz. 

29. 

1.2  d. 

25. 

1.728  cu.  ft. 

324 


ANSWERS. 


114-ise 


Page  114. 

24. 

174.5J  cu.  ft. 

12. 

^  rd. 

25. 

930.24  gal. 

13. 

x¥(f  bu- 

26. 

640  A. 

14. 

f  lb. 

27. 

5760  A. 

15. 

.001+  c.  yr. 

28. 

$3640. 

16. 

.6125  bu. 

29. 

$20.50. 

17. 

.66f  <£. 

Page  124. 

Page  115. 

18. 

643.5  liters. 

18. 

.3375  A. 

19. 

6070.5  meters. 

19. 

.7  lb. 

20. 

6040.08  grams. 

20. 

.18+  mi. 

21. 

23456  grams. 

21. 

23 

2  8* 

22. 

3.458  grams. 

22. 

tV 

23. 

45060  liters. 

23. 

24. 

3540  liters. 

24. 

3 

16* 

25. 

8450  sq.  meters. 

Page  117. 

26. 

1.324  meters. 

11. 

85£  yd. 

27. 

24  meters. 

12. 

22\  A. 

28. 

4.345  liters. 

13. 

$241920. 

29. 

322500  grams. 

14. 

/  1340|  boards. 

30. 

7.4635  kilograms. 

^  1345,  in  practice. 

31. 

240.59  yards. 

15. 

$9o.l85+. 

32. 

27.65+  miles. 

Page  118. 

33. 

9.4488  inches. 

16. 

7722  bricks. 

34. 

709.375  bushels. 

17. 

56  rings,  with  .8  pwt.  R. 

35. 

9.9+  gallons. 

18. 

75  reams. 

36. 

330.69  pounds. 

19. 

110.65+P. 

37. 

991.872  liters. 

20. 

7114  P. 

38. 

72.49  steres. 

21. 

$222,962+. 

39. 

30.35+ars. 

22. 

90  bu. 

40. 

536.448  meters. 

23. 

4.417875  A. 

41. 

107.07  sq.  meters. 

COMPOUND 

NUMBERS. 

I’age  125.  Page  126. 

1.  3T.  lOcwt.  15  lb.  1  oz.  lljfi-  dr.  9.  5  mi.  5  fur.  17  rd.  4  yd. 

2.  58  mi.  3  fur.  9  rd.  2  ft.  3T72  in.  10.  16  rd.  4  yd.  9  in. 

3.  36  w.  22  li.  11  min.  47  sec.  11.  20°  27/  44". 


4.  101  lb.  7  oz.  12  pwt.  12  gr. 

5.  131  bu.  1  pk.  1.  qt.  1  pt. 

6.  1  C.  11  S.  15°  54'  24". 

7.  18  cd.  54  cu.  ft. 


12.  51°  22'  6". 

13.  375  A.  3  R.  28  P. 

14.  5  yr.  3  mo.  23  da. 

1 5.  27  yr.  6  mo.  24  da. 


*  Revised  Problem. 

325 


126-139 


COMPLETE  ARITHMETIC. 


16.  93°. 

17.  15°31'N. 

Page  128. 

2.  2  fur.  35  rd.  2  yd.  1  ft.  6  in. 

3.  2248  bu.  1  pk.  2  qt. 

4.  60  T.  16  cwt.  28  lb. 

5.  2  lb.  8  oz.  3  pwt. 

6.  2 mi. 6 fur.  14 rd. lyd. 2 ft. 8 fin. 

7.  2  mi.  2  fur.  12  rd.  2  ft.  5  in. 

8.  l°52'l|f". 

9.  1  lb.  7  oz.  15  pwt.  17t9^  gr. 

10.  35  bu.  3  pk.  4  qt. 

Page  129. 

11.  3  da.  2  h.  51  min.  40  sec. 

12.  21  rd.  4  yd.  1  ft.  3r6T  in. 

13.  10  lb.  10  oz.  lOf  dr. 

14.  95  rings,  with  15  gr.  R. 

15.  12  kegs. 

16.  264  rotations. 

17.  120  lengths. 

18.  48  barrels. 

19.  627  axes,  with  1  lb.  7  oz.  R. 


20. 

1188  steps. 

21. 

213  da.  1  li.  23  min.  28  sec. 

Page  133. 

16. 

22  min.  4  sec. 

17. 

27  min.  59  sec.  past  9  o’clock 

A.  M. 

18. 

9  min.  5  sec.  past  3  P.  M. 

19. 

34  min.  31  f  sec.  past  3  P. 

M. 

20. 

37  min.  52r25  sec.  past  5  P. 

M. 

21. 

23°  48'. 

22. 

33°  47'  30". 

23. 

40°  15'  W. 

24. 

97°  2'  30"  W. 

25. 

53*f  min.  past  10  A.  M. 

26. 

52 1  min.  past  9  A.  M. 

Page  134. 

27. 

1  h.  42  min. 

28. 

31  min.  16  sec. 

29. 

1  li.  6  min.  3§  sec. 

30. 

2  h.  32  min.  1  sec. 

31. 

3  h.  13  min.  45  sec. 

32. 

5  h.  37  min.  52T2^  sec. 

Page  136. 

12.  .09 

13.  45 

14.  2.20 

1 5.  .24f,  or  .244 

16.  .30*,  or  .3025 

17.  .ooxV 

1 8.  .20*,  or  .2025 

Page  138. 

6.  12.25 

7.  32.4 

8.  180. 

9.  9.375 

10.  $49.50 

11.  $3. 

12.  $5.4075 


PERCENTAGE. 

13.  141b. 

14.  1.52  lb. 

15.  70  days. 

16.  $2.70 

17.  $4. 

18.  93  ft. 

19.  $.36375 

20.  321f  days. 

21.  $.09 

22.  -135 

23.  If 

24.  .054 

25.  .046875 

26.  36  miles. 

27.  2618  lb. 

28.  28 1  tons. 


29.  $206.25 

30.  By  R.  R.,  17062.5 
bushels. 

Page  139. 

7.  6% 

8.  20% 

9.  33*% 

10.  20% 

11.  16f  % 

12.  2 *% 

13.  *% 

14.  7 *% 

15.  22% 

16.  6% 

17.  13*% 

18.  20% 


326 


ANSWERS. 


139-157 


12.  7920  pounds. 

13.  $182.50 

14.  $80.60 

15.  525  pupils. 

16.  14500. 

17.  $50. 

18.  $14400. 


19.  75% 

Page  145. 

16  f  $38.59+ 

20.  40% 

144  acres. 

1  $578.91— 

21.  83  J% 

17. 

122.4  “ 

1  7.  $76312.50 

22.  30% 

80  “ 

18.  $1950. 

23.  15% 

_  133.6  “ 

Page  152. 

24.  91f  % 

18.  60% 

19.  12% 

25.  55*  % 

19.  36y6g. 

20.  6f  % 

Page  140. 

20.  $700. 

21.  $30000. 

26.  78f% 

21.  $6480. 

22.  $4170. 

27.  45% 

22.  $9500. 

23.  $5948.25 

28.  20lfo 

23.  $14175. 

24.  $3460. 

Page  141. 

Apple,  540. 

25.  $11600.  C’n  $174. 

7.  $15200. 

24 

Peach,  264. 

26.  $4.80 

8.  731f. 

Cherry,  150. 

27.  $39. 

9.  800. 

^Pear,  246. 

28.  $113.01  + 

10.  716f. 

25.  C 

’s  share,  $7175. 

29.  $1357.886 

1 1.  1500  sheep. 

26.  $24570 

30.  4434.589+bu. 

27.  37760. 

28.  60000. 

Page  148. 

19.  $7593.75 

20.  $29.70 

21.  20% 

22.  30% 


Page  153. 

31.  25%  of  sales 

32.  $12857.14+ 

33.  $2077.82+ 
f  25256.9+  lb. 


34. 


1  $403.06+ 

Page  156. 


19.  375  barrels. 

20.  $40000. 

21.  150000000  sq. 

22.  80000. 

Page  143. 

12.  $581.81* 

13.  240. 

14.  125. 

15.  2430. 

16.  450  acres. 

17.  $120. 


23.  14*% 

24.  12||  % 
i.  25.  16f% 

26.  $65280. 

27.  $2.50 

28.  $216. 

29.  $6750. 

30.  $36375. 

31.  $360. 

32.  14f  % 

33.  $4,735 


11.  $6937.50 

12.  $300. 

13.  $507.50 

Page  157. 

14.  $456. 

15.  40  shares. 

16.  $270. 

17.  $600. 

18.  $22000. 

19.  $8500. 

20.  8.48+% 


18.  $.66f 

19.  600  pupils. 

20.  800. 

21.  145400. 

Page  144. 

16.  74422. 


Page  149. 

34.  Sold  at  cost. 
Page  151. 

13.  $103,275 

14.  $199.06^ 

15.  $88,136 


qi  /  Rate,  10 \\% 

'  1  Div.,  $469.36+ 

22. 

23.  80  shares. 

24.  100  shares. 

25.  $7200  worth. 


327 


158-171 


COMPLETE  ARITHMETIC. 


Page  158. 

26.  $252. 

27.  50  shares. 

28.  H% 

Page  160. 

8.  $625. 

9.  $120,875 

Page  161. 

10.  $36.94— 

11.  2*. 

12.  f. 

13.  $15600. 

14.  $35000. 

15.  $495. 

16.  $840. 

j  1st  Co.,  $20000. 

17.  |  2d  Co.,  $24000. 
I 3d  Co.,  $16000. 

18.  $3290. 

19.  $20000. 

20.  $32000. 

Page  162. 

21.  $22500. 

22.  $12040. 

23.  $12000. 

24.  $2600. 

25.  $34650. 

26.  $48636. 

Page  163. 

27.  $111.90 

28.  $141.60 

29.  $2225. 

30.  $1250. 

Page  164. 

31.  $6958.40 

Page  165. 

2.  1£%,  or  12  mills. 

3.  Ito  %,  °r  13  mills. 
5  f  $12845.487, 

'  \  or  $12861.368 


6.  $130280.51+ 
Page  166. 

7.  $478747.83 

8.  $791131.56+ 
r  A’s  $203.40 

9.  \  B’s  $147,759 
l  C’s  $750.90 

10  J  6%,  $69.80 
13%,  $41.88 

11.  $96. 

12.  $258.16+ 

Page  167. 

14.  $10.62 

15.  $14.40 

16.  $1-287 

17.  $51,075 

18.  -$12.9075 

19.  $81.12 

20.  $1284. 

21.  $1363.62 

22.  $2255.10 

23.  $3380.25 

Page  169. 

1.  $328.90 

2.  $33480. 

3.  $4037.50 

4.  $7660.80 

5.  $3990.87+ 

6.  $22680. 

7.  $3589.525 
g  f  $2.20 

’  1  $3,696 
Page  170. 

1.  62% 

2.  Last,  $1385.70 
Q  f  05% 

‘  I  $1375. 

Page  171. 

4.  64|  cents. 


328 


ANSWERS. 


173-185 


INTEREST. 


Page  173. 

20.  $225,596 

21.  $6.53 

22.  $314,951 

23.  $79.08£ 

24.  $4,462 

Page  174. 

25.  $13,144 

26.  $16.78 

27.  $80,336 

28.  $136.81 

29.  $18.48 

30.  $173,919 

31.  $126,939 

32.  $807,202 

33.  $2,237 

34.  $166.31 

35.  $93,634 

37.  $23,869 

38.  $6516.661 

39.  $90,782 

40.  $155,564 

41.  $238,565 

42.  $187,716 

43.  $297,498 

44.  $242,804 

Page  175* 

45.  $66.54 

46.  $4,385 

47.  $814,123 

Page  178. 

29.  $117,072 

30.  $38.88 

31.  $41,483 

32.  $13,105 

33.  $1120.759 

34.  $77.54+ 

35.  $891,377 


36.  $119,523 

37.  $2238.944 

38.  $8,206 

39.  $19,188 

40.  $58.49+ 

41.  $286,888 

42.  $31,711 

43.  $53,946 

44.  $410.84 

45.  $169,928 

46.  $1440.555 

47.  $101,767 

48.  $76,611 

Page  180. 

2.  $3,075 

3.  $25,744 

4.  $67,225 

5.  $77,643 

6.  $123.33 

7.  $427,653 

8.  $153,208 

9.  $82,809 

10.  $152.40 

11.  $507,097 
Page  183. 

2.  $508,717 

3.  $170,151 

4.  $283,103 

5.  $293,147 

Page  184. 

6.  $463,761 

7.  $526,335 

8.  $210,806 

Page  185. 

9.  $436,923 

10.  $125,496 
1.  $3,906 


C.  Ar.  28.  329 


186-195 


COMPLETE  ARITHMETIC. 


Page  186. 

2.  $32,682 

3.  Ain’t,  $56,729 

4.  $86,458 

5.  $229,858 

6.  $2,354  (by  days.) 

7.  $952.66  “ 

8.  $7,791  “ 

9.  7% 

10.  5% 

11.  10% 

Page  187. 

12.  8% 

13.  8% 

14.  8% 

15.  8% 

16.  7% 

18.  2  yr.  6mo. 

19.  3  yr.  11  mo. 

20.  2  yr.  7  mo.  15  da. 

Page  188. 

21.  1  yr.  10  mo. 

22.  3  yr.  3  mo.  9  da. 

23.  10  yr. 

24.  8J  yr. 

25.  33 £  yr. 

Page  189. 

27.  $317.50 

28.  $3600. 

29.  $9000. 

30.  $5400. 

31.  $20000. 

32.  $7500. 

33.  $2540. 

Page  190. 

35.  $64,244 

36.  $540. 

37.  $360. 

38.  $2400. 

Page  191. 

1,  $35,636 


2.  $178.88 
*  3.  7% 

*4.  8% 

5.  2  yr.  9  mo.  23  da. 

6.  1  mo.  8  da. 

7.  6  mo.  24  da. 

8.  $408. 

9.  $35.60 

10.  $630. 

11.  $192.78 

12.  $600. 

13.  $265. 

14.  $383,195  (by  days.) 

15.  10  years. 

16.  $445,783+ 

17.  $14285.714 

Page  192. 

1.  $12  (dis.) 

2.  $10. 

Page  193. 

3.  $20,905 

4.  $7,722 

5.  $19,736 

6.  $94,821 

7.  $25,937 

8.  $20,203 

9.  $59,125 

10.  $4,766 

11.  $7,927 

12.  $20,216 
13.  $1.62+ 

14.  $4,233 

15.  $7,863 

16.  $.61+ 

17.  $400. 

18.  $5. 

Page  195. 

2.  $244,833  (proceeds.) 

3.  $143,223 

4.  $79,909+ 

5.  $980,625 


Revised  Problem. 

330 


ANSWERS. 


195-211 


6.  $741. 

7.  $1228.54+ 

8.  $54,852 
9  $117,973 

10.  $491.50 

1 1.  $8.50+ 

Page  196. 

12.  $391,944 

1 3.  $.29+ 

Note. — The  90  days  include  days 
of  grace. 

14  $.43+ 

15.  $.25+ 


6.  $644.15 

7.  $314,613 

8.  $256. 

9.  $980. 

10.  $1250. 

1 1.  $500. 

12.  $800. 

13.  $360. 

Page  204. 

1.  $1642.50 

2.  $1425. 

3.  $4500. 

4.  $616,197 


16.  July  12. 


Page  205. 


17.  April  9. 

18.  December  12. 

19.  $461,317 

20.  $124,242 

21.  $21,488 

22.  $90.20+ 

Page  197. 

23.  $379,482+ 


5.  $9775. 

6.  $1250. 

7.  $456,524 

8.  $958.33 J 

9.  $490,909 
10.  $.888 

11.  $273,125 

12.  7i% 


Note.— The  amount  due  April*13.  (1)  6;  (2)  5|;  (3)  5. 
4, 1870,  is  found  by  the  “  Merchant’s  Page  206. 


Rule”  (p.  184). 

24.  $500. 

25.  $2000 

26.  $836  729 

27.  $2000. 

28.  $725 

29.  $319  60  (no  grace.) 

30.  $978.50 

31.  $878.75 


2.  $966.72 

3.  $1298.80 

Page  207. 

4.  $4415.60 

5.  $770.12  (no  grace.) 

6.  $807,066 

7.  $544,956 

8.  $316,425 

Page  209. 


32  -f  $88*802+,  gain.  [grace, 
t  N.  B. — The  92  days  include 
Page  202. 

2.  $1243.75 

3.  $1052.625 

4.  $2537.50 

Page  203. 

5.  $502.25 


2.  $129,303 

3.  $389,568 

4.  $900,407 

5.  $141,191 

Page  211. 

6.  $1797  418 

7.  $800,516 

8.  $2074.296 


*  Revised  Problem. 


331 


212-226 


COMPLETE  ARITHMETIC. 


Page  212. 

16. 

6  months  after  maturity. 

2. 

7  months. 

17. 

£  U  U  ll 

3. 

8  months. 

Page  215. 

4. 

4  months. 

18. 

$400. 

5. 

7  months. 

19. 

10^  months  after  maturity. 

6. 

5Ty  months. 

20. 

Nov.  1,  1870. 

7. 

4  months. 

1. 

July  15,  1870. 

8. 

49  days  (48.9) 

Page  216. 

9. 

46  days. 

2. 

Oct.  14,  1869. 

Page  213. 

3. 

Oct.  9,  1870. 

10. 

68  days. 

4 

f  Aug.  30,  1868. 

11. 

46  days. 

f  $949.49  (at  6%). 

12. 

July  4,  1870. 

Page  219. 

13. 

July  29,  1870. 

8. 

March  6,  1870. 

14. 

July  6,  1870. 

9. 

Nov.  6,  1870. 

Page  214. 

10. 

Oct.  6,  1870. 

15. 

6  months  after  maturity. 

RATIO  AND 

PROPORTION. 

Page  221. 

34. 

3  :  7. 

13. 

35  :  17. 

35. 

7  :  13. 

14. 

3.4  :  .62. 

36. 

13  :  5. 

15. 

2*  :  f 

37. 

16  :  7. 

16. 

%b 

38. 

15  :  8. 

17. 

2f. 

39. 

10  :  9. 

18. 

3 

z- 

40. 

8  :  15. 

19. 

2.2 

41. 

10  :  11. 

20. 

.45 

42. 

14  :  15. 

21. 

2 

Z' 

43. 

14  :  15. 

22. 

9. 

44. 

33  :  28. 

23. 

J50 

1  2T* 

45. 

52  :  51. 

24. 

6. 

46. 

7  :  80. 

25. 

b 

47. 

14  :  5. 

26. 

b 

48. 

56  :  21. 

27. 

b 

49. 

4  :  9,  or  f. 

28. 

fV 

Page  222. 

29. 

2  :  5. 

50. 

36  :  55,  or  |f. 

30. 

5  :  12. 

51. 

4  :  7,  or  f. 

31. 

7  :  12. 

52. 

8  :  35,  or 

32. 

11  :  20. 

Page  226. 

33. 

3  :  4. 

14. 

12. 

332 


ANSWERS. 


226-236 


15.  48. 

16.  6. 

17.  30. 

18.  36. 

19.  .13 

20.  14.4 

21.  .75 

22.  xf. 

23.  ff. 

24.  ljf. 

25.  xV 

26.  54  lb. 

27.  14  oz. 

28.  90  days. 

29.  1. 

Page  227. 

31.  $1757.777 

32.  $56.25 

33.  $8571.43— 

34.  180  acres. 

35.  $6.90 

36.  36000. 

37.  66600. 

38.  34°  40'. 

39.  208  bbl.  (52  w.) 

40.  54  bbl. 

4 1.  200  ft. 

42.  150  ft. 

Page  228c 

43.  $6075. 

44.  $39.60 

45.  18£  tons. 

46.  96  apples. 

47.  41f  acres. 

49.  75  days. 

50.  36  men. 

5 1.  40  and  50. 

52.  288  and  352. 

Page  229. 

53.  $4500;  $5100. 

54.  60  miles;  90  miles. 


Page  230. 

2.  2  :  3. 

3.  25  :  27. 

4.  10  :  9. 

5.  16  :  45. 

6.  16  :  15  ::  16  :  15. 

Page  231. 

7.  7  :  5  ::  42  :  30. 

8.  3  :  14  ::  12  :  56. 

9.  $108  :  $216  ::  1  :  2. 
10o  10  :  9  ::  10  :  9. 

11.  39. 

12.  10. 

13.  33£. 

14.  81. 

15.  49. 

Page  232. 

16.  3  men. 

17.  18  men. 

18.  $60. 

19.  960  cu.  ft. 

Page  233. 

20.  63  bushels. 

2 1.  256  miles. 

22.  125  slabs. 

23.  $22.75 

24.  $99. 

25.  $675. 

26.  8  men. 

27.  10  days. 

28.  6  men. 

Page  236. 

o  f  A’s  $2500. 

\  B’s  $2000. 
r  A’s  $1600. 

3.  \  B’s  $  900. 

I  C’s  $  700. 
r  A’s  $2800. 

4.  \  B’s  $2100. 

I  C’s  $1400. 

5.  $4400  and  $3300. 


C.Ar.— 28. 


333 


236-247 


COMPLETE  ARITHMETIC. 


r  A’s  $5000. 
*  6.  Proceeds  -j  B’s  $8000. 

I  C’s  $7000. 


Page  238. 


8.  { 

{iVS  qpl 

B’s  $2 
C’s  $1 


A’s  $3175. 

B’s  $5715. 

A’s  $1700. 

$2300. 

$1000. 
r  A’s  $1350. 

10.  \  B’s  $1200. 

I  C’s  $1050. 

Page  243. 

g.  /  Younger,  $5985. 
bL  t  Elder,  $8550. 
go  f  Younger,  $2896. 

l  Elder,  $3258. 
gq  /  $12040. 

I  $15480. 

64.  $3240  and  $4860. 

65.  A,  $387.50;  B,  $310. 

66.  7220. 


Page  244. 

67.  19800  steps. 

68.  $620. 

69.  $212.50 


70.  $289.20 

71.  $857.50 

72.  $3756. 

73.  $131.25 

74.  $5654.40 

75.  $949,218 

76.  $137.50 

7  7.  20  days. 

78.  180  cords. 

Page  245. 

79.  15  days. 

g0  f  A’s  $1250. 
0U<  l  B’s  $2000. 

8  .  f  A’s  $1250. 

0  A*  l  B’s  $  900. 

r  A’s  $2550. 
82.  \  B’s  $1275. 

I  C’s  $  850. 
g3  f  A’s  $2502.50 
l  B’s  $2957.50 
g4  fA’s$1577f 
I  B’s  $  822f. 
gK  f  A’s  $900. 

I  B’s  $850. 

86.  56  days. 

{1st,  50  yd. 
2d,  45  yd. 
3d,  40  yd. 


INVOLUTION  AND  EVOLUTION. 


Page  246. 

7.  164836. 

8.  74088. 

9.  331776. 

10.  1048576. 

11.  42.25 

12.  .074088 


13.  .07776 

U  58  32 

•  TTnnnr* 

1  K  625 
10.  T^2p 

ID  14641 

10.  70  73^- 


Page  247. 

17.  388129. 

*  Revised  Problem. 

334 


ANSWERS. 


247-260 


18.  1157625. 

19.  1197.16 

20.  .000004096 

21.  3.8416 

22.  .002025 

23. 

24. 

25.  277*. 

26.  123ff 

27.  244fc. 

OQ  256 

aO.  TO??* 

29.  .000000000003125 

30.  8.489664 

31. 


219  7 
3  3  7¥* 
2401 
65ST* 


45.  166375. 

46.  512000. 


17.  87. 

18.  365. 

19.  459. 

20.  648. 

21.  702. 


Page  254. 


22.  6400. 

23.  3.24 

24.  13.3 

25.  .25 

26.  .094 

27.  24.221+ 

28.  2.343+ 

29.  116.047+ 


Page  248. 

30. 

25 

32. 

2809. 

31. 

32b 

33. 

2025. 

32. 

2.236 

34. 

5625. 

Page  259. 

35. 

1444. 

36. 

15625. 

11. 

35. 

(2).  45. 

37. 

15376. 

12. 

49. 

(2).  57. 

13. 

65. 

(2).  74. 

Page  249. 

14. 

89. 

(2).  97. 

15. 

364. 

38. 

148877. 

16. 

145. 

(2).  325. 

39. 

91125. 

17. 

345. 

(2).  352. 

40. 

32768. 

18. 

301. 

(2).  802. 

41. 

79507. 

19. 

57. 

(2).  504. 

42. 

157464. 

20. 

47. 

(2).  3002. 

43. 

42875. 

21. 

2.5 

(2).  .4203+ 

44. 

140608. 

22. 

4.6 

(2).  .25 

Page  260. 


Page  253. 

13.  58. 

14.  65. 

15.  88. 

16.  79. 


23.  3.83+  (2).  2.31+ 

24.  .035  (2).  4.08 

25.  2.08+  (2).  3.141+  (3). 

3.683+ 

26.  1.259+  (2).  2.714+  (3). 

5.846+ 


335 


260-267 


COMPLETE  ARITHMETIC. 


27.  A,  or*.  (2).  ft. 

28.  2*.  (2).  3J. 

29.  27  inches. 

30.  34.5  inches. 

3 1.  59.9+  inches. 

32.  24  feet. 

Page  264. 

1.  10. 

2.  12  inches 

3.  36  feet. 

Page  265. 

4.  240  yards. 

5.  30  feet. 

6.  120  rods. 

7.  225  feet. 

8.  119.73+  feet. 

9.  72.74+  rods. 

10.  104^  rods. 

Page  266. 

11.  176.715  sq.  in. 

12.  872f  sq.  yd. 


1 3.  10  yards. 

14.  64. 

15.  16. 

16.  558.5+  sq.  yd. 

17.  25  feet. 

18.  127.32+ 

19.  314.16  sq.  in. 

20.  196663355.75  sq.  m. 

Page  267. 

21.  259333411782.86 

22.  179.594  cu.  in. 

23.  64  balls. 

24.  64. 

25.  1000. 

26.  3000  miles. 

27.  33000  miles. 

28.  12  inches. 

29.  9.5  inches. 

30.  35.014+  feet. 

31.  70688.827  cu.  ft. 

32.  65512.7+  miles. 

33.  105434.8+  miles. 

34.  Moon’s  surface  about  TV  of 
the  earth’s  surface. 


Note. — Arithmeticians  are  not  agreed  respecting  the  order  in 
which  the  operations,  indicated  by  X  and  -j-  in  example  11th,  page 
88,  are  to  be  performed.  They  may  be  performed  in  their  order, 
from  left  to  right,  or  .08^  X  1-2J  may  be  divided  by  .006^  X  -016. 
The  first  method  is  preferred.  All  ambiguity  may  be  removed  by 
the  use  of  the  parenthesis.  If  written  thus:  (.085-  X  1.2£-f-  .006|-) 
X-016,  the  product  of  .08£X1.2J  is  divided  by  .006^  and  the  quo¬ 
tient  multiplied  by  .016.  If  written  thus:  (.08|X  1.2?)  -s-  (.006£X 
.016),  the  product  of  .08|-  X  1.2$  is  divided  by  the  product  of  .006^ 
X  *016.  See  Manual,  page  66. 


336 


ANSWERS. 


273-275 


GENERAL  REVIEW  PROBLEMS. 


Page  273. 

76.  658. 

77.  2997. 

78.  12. 

79.  $15.98 

80.  145§§  A. 

8 1.  m- 

82.  9*. 

8  3.  l^V 

84.  f. 

85.  5^>x. 

86.  251*. 

87.  *V 

88.  2|. 

89.  Xf« 

90.  $960. 

q  *  f  Value,  $32000. 
1  Part  left, 


Page  274. 

92.  7.039 

93.  .0301965 

94.  1024. 

95.  .002235 

96.  16000. 

97.  .075 

98.  .003 

99.  .000008 

100.  7  oz.  10  pwt. 

101.  .01 

C.  Ar.— 29. 


102.  $.14175. 

103.  41760  min. 

104.  7948800  sec. 

105.  509 If  steps. 

106.  422§. 

107.  18  A. 

108.  41^  yd. 

109.  184800  gr. 

1 10.  300  sq.  ft. 

111.  45  cts. 

112.  $15.20 


Page  275. 

1 13.  8  ft. 

114.  $72,875 

115.  130,  with  1  bu.  1  pk.  R. 

116.  $60.25 

117.  6415^. 

1 18.  28  min.  44  sec.  past  11  A.  M. 

119.  5  P.  M. 

120.  42  min.  51f  sec.  past  4  P.  M. 

121.  I 

122.  .65 

123.  jtzj' 

124.  tVst  16. 

125. 

126.  900  men. 

127.  llf% 

128.  $4.50 

129.  Gained 
337 


276-280 


COMPLETE  ARITHMETIC. 


Page  276. 


130.  4%  loss. 

131.  Lost,  $20. 

132.  $100. 

iqo  /  $986. 

1  Gained,  17  % . 

1 34.  $6  per  yard. 

135.  60% 

136.  0% 

137.  25% 

138.  $141,382 

139.  16ft% 

140.  $16.20 
141.  $2048. 


Page  277. 


142.  487804.87  lb. 

143.  400  yd. 

144.  $5,368 

145.  $11,111 

146.  $6,511 

147.  4i% 

148.  6H. 


149.  $466f. 


150. 


r  $600. 
*-8£% 
151.  $289,532 


152.  Discount,  $220. 


153.  $36. 

154.  $2,583 

155.  $.314 

156.  $310.61 

157.  9  months. 


Page  278. 

158.  6  months. 

159.  In  6|  months. 

160.  $75. 

161.  $11  and  $16. 


162.  24  men. 

163.  126  bushels. 


164.  $216. 

165.  $1. 

166.  $4800. 

167.  $2400. 

168.  Profits, 

169.  20.5 

170.  3.5 

171.  40  feet. 


f  A,  $555§. 
I B,  $333f. 


Page  279. 


172. 

65  miles. 

173. 

50  feet. 

174. 

160  rods. 

175. 

23. 3  — l —  feet. 

176. 

28.28+  feet. 

177. 

154  cu.  ft. 

178. 

78  sq.  ft. 

179. 

Area,  7.854  A. 

180. 

24.4+  in. 

181. 

64  balls. 

182. 

1728  blocks. 

183. 

128.57+  bu. 

184. 

3456  gal. 

185. 

564.019  gal. 

r  A,  $445. 

186. 

\  B,  $230. 

1C  $325. 

QO 

<1 

e 

6  days. 

Page  280. 

188. 

$545,454 

189. 

4J  miles. 

r  1st,  $3250. 

190. 

|  2d,  $3900. 

1 3d,  $1950. 

191. 

23|xi  bu. 

338 


ANSWERS, 


c  W.,  18 

i  ne  /  A  in  5  days. 

192.  \  M.,  22. 

1  Bin  4  days. 

lCh.,50. 

196.  2|  times. 

r  1st,  $240 

i  q  7  J  1st,  90  miles. 

193.  \  2d,  $180. 

1 2d,  40  miles. 

Ud,  $210. 

198.  7£  feet. 

c  A,  14§  days. 

194.  \  B,  72  “ 

Page  281. 

l  C,  lOf  “ 

199.  50  feet. 

200.  3  hours. 

APPENDIX. 

Page  287. 

Page  302. 

280-305 


1.  Fifty.  1.  160. 

2.  One  hundred  and  ninety-  2.  3645. 


five. 

3.  130. 

4.  400. 

5.  1300. 


Page  299. 

1.  23. 

2.  25. 

3.  45. 


3.  3* 

4.  b 

5  f  4th,  127 f. 

'  1 3d,  1827Hf. 

6.  3. 

7.  3640. 

g  f  $256. 

I  $511.50 
9.  $76293945.31 


Page  303. 


Page  300. 

4.  4. 

5.  2  b 

6.  25,  29,  33,  37. 

7.  34. 

8.  3870. 

g  f  6th,  225. 

*  \  7th,  3213. 

10.  156. 

22  /  Last  yard,  $1.49 
’  i  Trench,  $37.75 


2.  $.326 


Page  305. 

4.  1  lb.  of  each.  (1  ans.) 

{300  lb.  at  22  cents. 
200  lb.  at  28  “ 

500  1b.  at  30  “ 
c  Rye,  270  bu. 

6.  j  Barley,  60  bu. 
v  Oats,  60  bu. 


339 


305-30S 


COMPLETE  ARITHMETIC. 


{10f  pwt.  16  carat. 
10f  “  18  “ 

32 1  “  22  “ 

r  50  lb.  at  15  cents. 
8.  |  50  “  17  “ 

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UNIVERSITY  OF  ILLINOIS-URBANA 

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A  COMPLETE  ARITHMETIC  CINCINNATI 


POLITICAL  EC 


3  0112  017106888 

OMY. 


ANDREWS  S  MANUAL  OF  THE  CONSTITUTION. 

Manual  of  the  Constitution  of  the  T’nited  States.  iJe- 
signed  for  the  Instruction  of  American  Youth  in  the  Du¬ 
ties,  Obligations  and  Rights  of  Citizenship.  Bv  Israel 
Ward  Andrews,  D.  D.,  President  of  Marietta  College. 
121110,  cloth,  408  pp. 

While  the  primary  object  has  been  to  provide  a  suitable  text-book, 
a  conviction  that  a  knowledge  of  our  government  can  not  be  too  widely 
diffused,  and  that  large  numbers  would  welcome  a  good  book  on  this 
subject,  has  led  to  the  attempt  to  make  this  volume  a  manual  adapted 
for  consultation  and  reference,  as  well  for  citizens  at  large  as  for  stu¬ 
dents.  With  this  end  in  view  the  work  embodies  that  kind  uf  mforma- 
tion  on  the  various  topics  which  an  intelligent  citizen  would  desire  to 
possess.  - 

GREGORY  S  POLITICAL  ECONOMY. 

A  New  Political  Economy.  By  John  M.  Gregory, 
LL.D.,  late  President  III.  Industrial  University.  1 2mo, 

t  ^  7 

393  PP- 

An  essentially  new  statement  of  the  facts  and  principles  of  Political 
Economy,  in  the  following  particulars : 

x.  The  clear  recognition  of  the  three  great  economic  facts  of  Wants, 
Work  and  Wealth,  as  the  principal  and  constant  factors  of  the  indus¬ 
tries,  and  as  constituting,  therefore,  the  field  of  Ecc  nomic  Science. 

2.  The  recognition  of  man  and  of  the  two  great  crystallizations  of 
man  into  society  and  into  states,  as  presenting  three  distinct  fields  of 
Economic  Science,  each  having  its  own  set  of  problems,  and  each  its 
own  species  of  quantities  or  factors,  to  be  taken  into  account  in  the 
solution  of  problems. 

3.  A  new  definition  and  description  of  Value  as  made  up  of  its  three 
essential  and  ever-present  factors  forming  the  triangle  of  Value,  and 
evidenced  by  the  clear  explanation  they  afford  of  the  various  fluctua¬ 
tions  of  prices. 

4.  The  new  division  and-  distribution  of  the  discussion  arising,  out 
of  these  new  fundamental  facts  and  definitions. 

5.  The  aid  rendered  to  the  reader  and  student  by  the  diagrams  and 
synoptical  views. 


Van  Antwerp,  Bragg  &  Co.,  Publishers, 


CINCINNATI  and  NEW  YORK. 


